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If a matrix A is such that 3 A3 + 2 A2 + 5 A + I = 0, then A-1 is equal to- a)- (3 A2 + 2 A + 5I)
- b)3 A2 + 2 A + 5I
- c)- 3 A2 - 2 A - 5
- d)none of these
Correct answer is option 'C'. Can you explain this answer?
If a matrix A is such that 3 A3 + 2 A2 + 5 A + I = 0, then A-1 is equal to
a)
- (3 A2 + 2 A + 5I)
b)
3 A2 + 2 A + 5I
c)
- 3 A2 - 2 A - 5
d)
none of these
|
Vikram Kapoor answered |
3A3 + 2A2 + 5A + I = 0
3A3 + 2A2 + 5A + AA−1 = 0
A−1 = −3A2 − 2A − 5
3A3 + 2A2 + 5A + AA−1 = 0
A−1 = −3A2 − 2A − 5
Which of the following statements is/are incorrect?(i) Adjoint of a symmetric matrix is symmetric.
(ii) Adjoint of a unit matrix is a unit matrix.
(iii) A (adj a) = (adj A) A = |A|
(iv) Adjoint of a diagonal matrix is a diagonal matrix. - a)(i)
- b)(ii)
- c)(iii) and (iv)
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
Which of the following statements is/are incorrect?
(i) Adjoint of a symmetric matrix is symmetric.
(ii) Adjoint of a unit matrix is a unit matrix.
(iii) A (adj a) = (adj A) A = |A|
(iv) Adjoint of a diagonal matrix is a diagonal matrix.
(ii) Adjoint of a unit matrix is a unit matrix.
(iii) A (adj a) = (adj A) A = |A|
(iv) Adjoint of a diagonal matrix is a diagonal matrix.
a)
(i)
b)
(ii)
c)
(iii) and (iv)
d)
None of these
|
Arshiya Mehta answered |
d) None of these All of the statements are correct.
(i) Adjoint of a symmetric matrix is symmetric. This is true because the adjoint of any matrix is the transpose of its conjugate,and if the matrix is symmetric, its transpose is the same as its conjugate.
(ii) Adjoint of a unit matrix is a unit matrix.This is true because the adjoint of a unit matrix is simply its transpose, which is also a unit matrix.
(iii) A(adja) (adjA)A=|A|.This is true because the adjoint of any matrix is the transpose of its conjugate,so A(adja)=(adjA) A= the transpose of the conjugate of A times A,which is equal to the determinant of A.
(iv) Adjoint of a diagonal matrix is adiagonal matrix. This is also true because the adjoint of a diagonal matrix is simply its transpose, which is also diagonal.
(i) Adjoint of a symmetric matrix is symmetric. This is true because the adjoint of any matrix is the transpose of its conjugate,and if the matrix is symmetric, its transpose is the same as its conjugate.
(ii) Adjoint of a unit matrix is a unit matrix.This is true because the adjoint of a unit matrix is simply its transpose, which is also a unit matrix.
(iii) A(adja) (adjA)A=|A|.This is true because the adjoint of any matrix is the transpose of its conjugate,so A(adja)=(adjA) A= the transpose of the conjugate of A times A,which is equal to the determinant of A.
(iv) Adjoint of a diagonal matrix is adiagonal matrix. This is also true because the adjoint of a diagonal matrix is simply its transpose, which is also diagonal.
If A is 3 × 4 matrix and B is a matrix such that A'B and BA' are both defined, then the order of B is- a)3 × 4
- b)3 × 3
- c)4 × 4
- d)4 × 3
Correct answer is option 'A'. Can you explain this answer?
If A is 3 × 4 matrix and B is a matrix such that A'B and BA' are both defined, then the order of B is
a)
3 × 4
b)
3 × 3
c)
4 × 4
d)
4 × 3
|
Sahana Sharma answered |
I'm sorry, there seems to be a missing part of the question. Can you please provide more information or context?
A is a skew-symmetric matrix and n is a positive integer, then An is - a)a symmetric matrix
- b)a skew-symmetric matrix
- c)a diagonal matrix
- d)none of these
Correct answer is option 'D'. Can you explain this answer?
A is a skew-symmetric matrix and n is a positive integer, then An is
a)
a symmetric matrix
b)
a skew-symmetric matrix
c)
a diagonal matrix
d)
none of these
|
Sahil Kapoor answered |
Skew-Symmetric Matrix
A skew-symmetric matrix is a square matrix where the elements below the diagonal are the negatives of those above the diagonal.
Properties of Skew-Symmetric Matrix
- The diagonal elements are all zero.
- The transpose of a skew-symmetric matrix is its negative.
- The determinant of a skew-symmetric matrix is zero if the size of the matrix is odd.
An to the power of n
When we raise a matrix to the power of n, we multiply the matrix n times by itself.
An is a matrix obtained by multiplying the matrix A n times by itself.
Answer
If A is a skew-symmetric matrix and n is a positive integer, then An is not necessarily a symmetric or skew-symmetric matrix, nor is it necessarily a diagonal matrix. Therefore, the correct answer is option 'D' - none of these.
A skew-symmetric matrix is a square matrix where the elements below the diagonal are the negatives of those above the diagonal.
Properties of Skew-Symmetric Matrix
- The diagonal elements are all zero.
- The transpose of a skew-symmetric matrix is its negative.
- The determinant of a skew-symmetric matrix is zero if the size of the matrix is odd.
An to the power of n
When we raise a matrix to the power of n, we multiply the matrix n times by itself.
An is a matrix obtained by multiplying the matrix A n times by itself.
Answer
If A is a skew-symmetric matrix and n is a positive integer, then An is not necessarily a symmetric or skew-symmetric matrix, nor is it necessarily a diagonal matrix. Therefore, the correct answer is option 'D' - none of these.
If the rank of a matrix A is 2, then the rank of 2A is- a)4
- b)1
- c)2
- d)none of these
Correct answer is option 'C'. Can you explain this answer?
If the rank of a matrix A is 2, then the rank of 2A is
a)
4
b)
1
c)
2
d)
none of these
|
|
Shreya Chauhan answered |
Explanation:
Rank of a matrix represents the number of linearly independent rows or columns in the matrix. If the rank of a matrix A is 2, then it means that there are 2 linearly independent rows or columns in the matrix.
To find the rank of 2A, we need to consider the matrix 2A and perform row or column operations to bring it into row or column echelon form.
Let A be an m x n matrix, then 2A will also be an m x n matrix.
Suppose we perform row or column operations on 2A to bring it into row echelon form. Then the first row or column will be non-zero as the first row or column of A is non-zero. Therefore, the rank of 2A will be at least 1.
Now, let's consider the case where A is a 2 x 2 matrix with rank 2. Then, the two rows or columns of A are linearly independent. Let A = [a1 a2] where a1 and a2 are the two columns of A.
If we multiply A by 2, we get 2A = [2a1 2a2]. Since a1 and a2 are linearly independent, 2a1 and 2a2 are also linearly independent. Therefore, the rank of 2A is also 2.
Hence, the correct answer is option C.
Rank of a matrix represents the number of linearly independent rows or columns in the matrix. If the rank of a matrix A is 2, then it means that there are 2 linearly independent rows or columns in the matrix.
To find the rank of 2A, we need to consider the matrix 2A and perform row or column operations to bring it into row or column echelon form.
Let A be an m x n matrix, then 2A will also be an m x n matrix.
Suppose we perform row or column operations on 2A to bring it into row echelon form. Then the first row or column will be non-zero as the first row or column of A is non-zero. Therefore, the rank of 2A will be at least 1.
Now, let's consider the case where A is a 2 x 2 matrix with rank 2. Then, the two rows or columns of A are linearly independent. Let A = [a1 a2] where a1 and a2 are the two columns of A.
If we multiply A by 2, we get 2A = [2a1 2a2]. Since a1 and a2 are linearly independent, 2a1 and 2a2 are also linearly independent. Therefore, the rank of 2A is also 2.
Hence, the correct answer is option C.
The eigen values of matrix A = 
are- a)± 3√3
- b)3, 1
- c)- 1 ± 2 √7
- d)none of these
Correct answer is option 'C'. Can you explain this answer?
The eigen values of matrix A = are
area)
± 3√3
b)
3, 1
c)
- 1 ± 2 √7
d)
none of these
|
Santosh Kumar answered |
From the property of Eigen values of Matrix we know
1. sum of eigen values = Trace of the matrix.
Trace of the matrix is sum of Principal diagonal elements.
So sum of P.D.E = 3+(-5) =-2
2. Product of Eigen values is equal to the determinant of the Matrix.
So determinant of the matrix= -27
Only option C satisfies the above two results. so option C is the answer
1. sum of eigen values = Trace of the matrix.
Trace of the matrix is sum of Principal diagonal elements.
So sum of P.D.E = 3+(-5) =-2
2. Product of Eigen values is equal to the determinant of the Matrix.
So determinant of the matrix= -27
Only option C satisfies the above two results. so option C is the answer
If 5 is one of the roots of equation 
then other two roots of equation are- a)– 2, 7
- b)– 2, – 7
- c)2, 7
- d)2, – 7
Correct answer is option 'D'. Can you explain this answer?
If 5 is one of the roots of equation then other two roots of equation are
then other two roots of equation area)
– 2, 7
b)
– 2, – 7
c)
2, 7
d)
2, – 7
|
Pie Academy answered |
x[(x2 − (2 × 8))] − 3[2x − (−2x × 7)] + 7(16 − 7x) = 0
x3 + 16x − 42 − 6x + 112 − 49x = 0
x3 − 39x + 70 = 0
(x − 5)(x2 + 5 − 14) = 0
(x − 5)(x2 + 7x − 2x − 14) = 0
(x − 5)(x + 7)(x − 2)
x = −7, 2
Hence, the correct option is (D).
x3 + 16x − 42 − 6x + 112 − 49x = 0
x3 − 39x + 70 = 0
(x − 5)(x2 + 5 − 14) = 0
(x − 5)(x2 + 7x − 2x − 14) = 0
(x − 5)(x + 7)(x − 2)
x = −7, 2
Hence, the correct option is (D).
is singular, then λ equals
If B is a non-singular matrix and A is a square matrix, then det(B−1 AB) is equal to- a)det B
- b)det A
- c)det (B−1)
- d)det (A−1)
Correct answer is option 'B'. Can you explain this answer?
If B is a non-singular matrix and A is a square matrix, then det(B−1 AB) is equal to
a)
det B
b)
det A
c)
det (B−1)
d)
det (A−1)
|
Maitri Sen answered |
The determinant of a product of matrices is equal to the product of the determinants of the individual matrices. Therefore, if B is a non-singular matrix and A is a square matrix, then det(B*A) = det(B) * det(A).
Since B is non-singular, its determinant is non-zero, det(B) ≠ 0. Therefore, det(B*A) = det(B) * det(A) ≠ 0.
This means that the product matrix B*A is also non-singular.
Since B is non-singular, its determinant is non-zero, det(B) ≠ 0. Therefore, det(B*A) = det(B) * det(A) ≠ 0.
This means that the product matrix B*A is also non-singular.
If 3x + 2y + z = 0, x + 4y + z = 0, 2x + y + 4z = 0 be a system of equations, then- a)it is inconsistent
- b)it has only the trivial solution x = 0, y = 0, z = 0
- c)it can be reduced to a single equation and so a solution does not exist
- d)determinant of the matrix of coefficients is zero
Correct answer is option 'B'. Can you explain this answer?
If 3x + 2y + z = 0, x + 4y + z = 0, 2x + y + 4z = 0 be a system of equations, then
a)
it is inconsistent
b)
it has only the trivial solution x = 0, y = 0, z = 0
c)
it can be reduced to a single equation and so a solution does not exist
d)
determinant of the matrix of coefficients is zero
|
|
Chirag Nambiar answered |
Understanding the System of Equations
The given system of equations is:
1. 3x + 2y + z = 0
2. x + 4y + z = 0
3. 2x + y + 4z = 0
To analyze this system, we can represent it in matrix form and determine its solution.
Matrix Representation
The equations can be represented in matrix form as:
\[
\begin{bmatrix}
3 & 2 & 1 \\
1 & 4 & 1 \\
2 & 1 & 4
\end{bmatrix}
\begin{bmatrix}
x \\
y \\
z
\end{bmatrix}
=
\begin{bmatrix}
0 \\
0 \\
0
\end{bmatrix}
\]
Calculating the Determinant
To find if the system has only the trivial solution, we calculate the determinant of the coefficient matrix:
\[
\text{det} = 3(4 \cdot 4 - 1 \cdot 1) - 2(1 \cdot 4 - 1 \cdot 2) + 1(1 \cdot 1 - 4 \cdot 2)
\]
Calculating this, we find that the determinant is non-zero, indicating that the system has only the trivial solution.
Conclusion
Since the determinant is non-zero, the only solution to the equations is:
- x = 0
- y = 0
- z = 0
Thus, the correct answer is option 'B': it has only the trivial solution x = 0, y = 0, z = 0.
The given system of equations is:
1. 3x + 2y + z = 0
2. x + 4y + z = 0
3. 2x + y + 4z = 0
To analyze this system, we can represent it in matrix form and determine its solution.
Matrix Representation
The equations can be represented in matrix form as:
\[
\begin{bmatrix}
3 & 2 & 1 \\
1 & 4 & 1 \\
2 & 1 & 4
\end{bmatrix}
\begin{bmatrix}
x \\
y \\
z
\end{bmatrix}
=
\begin{bmatrix}
0 \\
0 \\
0
\end{bmatrix}
\]
Calculating the Determinant
To find if the system has only the trivial solution, we calculate the determinant of the coefficient matrix:
\[
\text{det} = 3(4 \cdot 4 - 1 \cdot 1) - 2(1 \cdot 4 - 1 \cdot 2) + 1(1 \cdot 1 - 4 \cdot 2)
\]
Calculating this, we find that the determinant is non-zero, indicating that the system has only the trivial solution.
Conclusion
Since the determinant is non-zero, the only solution to the equations is:
- x = 0
- y = 0
- z = 0
Thus, the correct answer is option 'B': it has only the trivial solution x = 0, y = 0, z = 0.
The inverse of the matrix 
is- a)
- b)
- c)
- d)
Correct answer is option 'C'. Can you explain this answer?
The inverse of the matrix is
isa)
b)
c)
d)
|
Veda Institute answered |
The matrix provided is:
To find the inverse of a diagonal matrix, we simply take the reciprocal of each non-zero diagonal element. The inverse of the matrix will be:
So, the inverse matrix is:
If A is a 3-rowed square matrix, then | 5 A | is equal to- a)5 | A |
- b)25 | A |
- c)125 | A |
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
If A is a 3-rowed square matrix, then | 5 A | is equal to
a)
5 | A |
b)
25 | A |
c)
125 | A |
d)
None of these
|
Sneha Menon answered |
To solve this problem, let's break it down into steps:
Step 1: Find the determinant of matrix A
The determinant of a square matrix is a scalar value that represents certain properties of the matrix. In this case, we need to find the determinant of matrix A.
Step 2: Multiply the determinant by 5
Once we have the determinant of matrix A, we need to multiply it by 5 to find | 5A |.
Step 3: Simplify the expression
Finally, we simplify the expression to determine the correct answer.
Let's go through each step in detail:
Step 1: Find the determinant of matrix A
The determinant of a 3x3 matrix can be calculated using the formula:
| A | = a(ei - fh) - b(di - fg) + c(dh - eg)
where a, b, c, d, e, f, g, h, and i represent the elements of matrix A.
Step 2: Multiply the determinant by 5
Now that we have the determinant of matrix A, we multiply it by 5:
| 5A | = 5 * | A |
Step 3: Simplify the expression
To simplify the expression, we need to evaluate the determinant of matrix A and then multiply it by 5.
Since we don't have the specific values of matrix A, we can't calculate the determinant. However, we can still determine the relationship between | 5A | and | A |.
When we multiply a matrix by a scalar, such as 5, the determinant is also multiplied by the same scalar. This property is known as the scalar multiple property of determinants.
Therefore, the correct answer is option C: 125 | A |.
Step 1: Find the determinant of matrix A
The determinant of a square matrix is a scalar value that represents certain properties of the matrix. In this case, we need to find the determinant of matrix A.
Step 2: Multiply the determinant by 5
Once we have the determinant of matrix A, we need to multiply it by 5 to find | 5A |.
Step 3: Simplify the expression
Finally, we simplify the expression to determine the correct answer.
Let's go through each step in detail:
Step 1: Find the determinant of matrix A
The determinant of a 3x3 matrix can be calculated using the formula:
| A | = a(ei - fh) - b(di - fg) + c(dh - eg)
where a, b, c, d, e, f, g, h, and i represent the elements of matrix A.
Step 2: Multiply the determinant by 5
Now that we have the determinant of matrix A, we multiply it by 5:
| 5A | = 5 * | A |
Step 3: Simplify the expression
To simplify the expression, we need to evaluate the determinant of matrix A and then multiply it by 5.
Since we don't have the specific values of matrix A, we can't calculate the determinant. However, we can still determine the relationship between | 5A | and | A |.
When we multiply a matrix by a scalar, such as 5, the determinant is also multiplied by the same scalar. This property is known as the scalar multiple property of determinants.
Therefore, the correct answer is option C: 125 | A |.
, the ratio of the cofactor to its minor of the element – 3 is
Study the following assertions about a square matrix
(i) The sum of the eigen values of A is equal to its trace
(ii) The product of the eigen values of A is equal to its determinant
(iii) All eigen values of A are non-zero, if and only if A is non-singular
(iv) If A-1 exists, then the eigen-values of A-1 are equal to the reciprocal of the eigenQ. Which of the following is correct with respect to above assertions?- a)Only (iii) and (iv) are true
- b)Only (i) and (ii) are true
- c)Only (ii), (iii) and (iv) are true
- d)(i), (ii), (iii), (iv) all are true
Correct answer is option 'D'. Can you explain this answer?
Study the following assertions about a square matrix
(i) The sum of the eigen values of A is equal to its trace
(ii) The product of the eigen values of A is equal to its determinant
(iii) All eigen values of A are non-zero, if and only if A is non-singular
(iv) If A-1 exists, then the eigen-values of A-1 are equal to the reciprocal of the eigen
(i) The sum of the eigen values of A is equal to its trace
(ii) The product of the eigen values of A is equal to its determinant
(iii) All eigen values of A are non-zero, if and only if A is non-singular
(iv) If A-1 exists, then the eigen-values of A-1 are equal to the reciprocal of the eigen
Q. Which of the following is correct with respect to above assertions?
a)
Only (iii) and (iv) are true
b)
Only (i) and (ii) are true
c)
Only (ii), (iii) and (iv) are true
d)
(i), (ii), (iii), (iv) all are true
|
|
Anirban Kapoor answered |
Assertion about Square Matrix
Sum of Eigen Values
The sum of eigenvalues of a square matrix is equal to the trace of the matrix.
Product of Eigen Values
The product of eigenvalues of a square matrix is equal to the determinant of the matrix.
Non-zero Eigen Values
All eigenvalues of a square matrix are non-zero, if and only if the matrix is non-singular.
Eigen Values of Inverse Matrix
If A-1 exists, then the eigenvalues of A-1 are equal to the reciprocal of the eigenvalues of A.
Correct Assertion
All the above assertions are correct.
Explanation
- The sum of eigenvalues of a matrix is equal to the trace of a matrix. This is because the trace of a matrix is equal to the sum of its diagonal elements, which are also the eigenvalues of the matrix.
- The product of eigenvalues of a matrix is equal to the determinant of a matrix. This is because the determinant of a matrix is equal to the product of its eigenvalues.
- All eigenvalues of a matrix are non-zero if and only if the matrix is non-singular. This is because a matrix is singular if and only if its determinant is zero, which means at least one of its eigenvalues is zero.
- If A-1 exists, then the eigenvalues of A-1 are equal to the reciprocal of the eigenvalues of A. This is because the eigenvalues of A-1 are the inverse of the eigenvalues of A.
Hence, all the given assertions are correct and option D is the correct answer.
Sum of Eigen Values
The sum of eigenvalues of a square matrix is equal to the trace of the matrix.
Product of Eigen Values
The product of eigenvalues of a square matrix is equal to the determinant of the matrix.
Non-zero Eigen Values
All eigenvalues of a square matrix are non-zero, if and only if the matrix is non-singular.
Eigen Values of Inverse Matrix
If A-1 exists, then the eigenvalues of A-1 are equal to the reciprocal of the eigenvalues of A.
Correct Assertion
All the above assertions are correct.
Explanation
- The sum of eigenvalues of a matrix is equal to the trace of a matrix. This is because the trace of a matrix is equal to the sum of its diagonal elements, which are also the eigenvalues of the matrix.
- The product of eigenvalues of a matrix is equal to the determinant of a matrix. This is because the determinant of a matrix is equal to the product of its eigenvalues.
- All eigenvalues of a matrix are non-zero if and only if the matrix is non-singular. This is because a matrix is singular if and only if its determinant is zero, which means at least one of its eigenvalues is zero.
- If A-1 exists, then the eigenvalues of A-1 are equal to the reciprocal of the eigenvalues of A. This is because the eigenvalues of A-1 are the inverse of the eigenvalues of A.
Hence, all the given assertions are correct and option D is the correct answer.
The rank of a unit matrix of order n is- a)1
- b)2
- c)0
- d)n
Correct answer is option 'D'. Can you explain this answer?
The rank of a unit matrix of order n is
a)
1
b)
2
c)
0
d)
n
|
|
Sagarika Yadav answered |
The rank of a matrix refers to the maximum number of linearly independent rows or columns in the matrix. In other words, it is the number of non-zero rows or columns in the matrix when it is in its row echelon form or reduced row echelon form.
The unit matrix, also known as the identity matrix, is a special type of matrix with ones on the main diagonal and zeros elsewhere. For example, the unit matrix of order 3 is:
```
1 0 0
0 1 0
0 0 1
```
To determine the rank of a unit matrix of order n, we need to find the maximum number of linearly independent rows or columns in the matrix.
Let's consider a unit matrix of order n:
```
1 0 0 ... 0
0 1 0 ... 0
0 0 1 ... 0
. . . ... .
. . . ... .
. . . ... .
0 0 0 ... 1
```
Since each row or column in a unit matrix is linearly independent from each other, the maximum number of linearly independent rows or columns is equal to the order of the matrix, which is n.
Therefore, the rank of a unit matrix of order n is n, as mentioned in option 'D'.
The unit matrix, also known as the identity matrix, is a special type of matrix with ones on the main diagonal and zeros elsewhere. For example, the unit matrix of order 3 is:
```
1 0 0
0 1 0
0 0 1
```
To determine the rank of a unit matrix of order n, we need to find the maximum number of linearly independent rows or columns in the matrix.
Let's consider a unit matrix of order n:
```
1 0 0 ... 0
0 1 0 ... 0
0 0 1 ... 0
. . . ... .
. . . ... .
. . . ... .
0 0 0 ... 1
```
Since each row or column in a unit matrix is linearly independent from each other, the maximum number of linearly independent rows or columns is equal to the order of the matrix, which is n.
Therefore, the rank of a unit matrix of order n is n, as mentioned in option 'D'.
For what value of λ, the system of equations x - 2y + z = 0, 2x - y + 3z = 0, x + y - z = 0 has the trivial solution as the only solution.- a)λ = - 4/5
- b)λ ≠ - 4/5
- c)λ ≠ 2
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
For what value of λ, the system of equations x - 2y + z = 0, 2x - y + 3z = 0, x + y - z = 0 has the trivial solution as the only solution.
a)
λ = - 4/5
b)
λ ≠ - 4/5
c)
λ ≠ 2
d)
None of these
|
|
Rohan Desai answered |
I'm sorry, but your question seems to be incomplete. Could you please provide more information?
If every minor of order r of a matrix A is zero, then rank of A is- a)greater than r
- b)equal to r
- c)less than or equal to r
- d)less than r
Correct answer is option 'D'. Can you explain this answer?
If every minor of order r of a matrix A is zero, then rank of A is
a)
greater than r
b)
equal to r
c)
less than or equal to r
d)
less than r
|
|
Mahi Dasgupta answered |
Explanation:
Given, every minor of order r of a matrix A is zero.
Let's first understand what is meant by minor of a matrix:
Minor of order r of a matrix A is the determinant of any square submatrix of A of size r x r.
Now, let's proceed to prove the given statement.
Proof:
Assume that rank(A) is greater than or equal to r.
Then, there exists a submatrix of A of order r that has a non-zero determinant.
This implies that the minor of order r of A is non-zero, which contradicts the given statement that every minor of order r of A is zero.
Hence, our assumption that rank(A) is greater than or equal to r is false.
Now, assume that rank(A) is less than r.
Then, there exists a linearly dependent set of r columns of A.
Without loss of generality, let these columns be the first r columns of A.
Then, any submatrix of A of order r formed by selecting any r rows and any r columns from the first r columns of A will have a determinant equal to zero, since the columns are linearly dependent.
This implies that every minor of order r of A is zero, which satisfies the given statement.
Hence, our assumption that rank(A) is less than r is true.
Therefore, the rank of A is less than r.
Hence, the correct answer is option 'D'.
Given, every minor of order r of a matrix A is zero.
Let's first understand what is meant by minor of a matrix:
Minor of order r of a matrix A is the determinant of any square submatrix of A of size r x r.
Now, let's proceed to prove the given statement.
Proof:
Assume that rank(A) is greater than or equal to r.
Then, there exists a submatrix of A of order r that has a non-zero determinant.
This implies that the minor of order r of A is non-zero, which contradicts the given statement that every minor of order r of A is zero.
Hence, our assumption that rank(A) is greater than or equal to r is false.
Now, assume that rank(A) is less than r.
Then, there exists a linearly dependent set of r columns of A.
Without loss of generality, let these columns be the first r columns of A.
Then, any submatrix of A of order r formed by selecting any r rows and any r columns from the first r columns of A will have a determinant equal to zero, since the columns are linearly dependent.
This implies that every minor of order r of A is zero, which satisfies the given statement.
Hence, our assumption that rank(A) is less than r is true.
Therefore, the rank of A is less than r.
Hence, the correct answer is option 'D'.
If A is a square matrix such that AA` = I, then value of A`A is- a)A2
- b)I
- c)A-1
- d)0
Correct answer is option 'B'. Can you explain this answer?
If A is a square matrix such that AA` = I, then value of A`A is
a)
A2
b)
I
c)
A-1
d)
0
|
Ishika Singhania answered |
To find the value of A`A, we need to understand the given information and use some properties of matrices.
Given: A is a square matrix such that AA` = I.
Understanding the given information:
1. A is a square matrix: A square matrix is a matrix in which the number of rows is equal to the number of columns. Let's assume A is an n x n matrix.
2. AA` = I: Here, AA` represents the product of matrix A with its transpose A`. The result of this product is the identity matrix I.
Using properties of matrices:
1. Transpose of a product: (AB)` = B`A`
- The transpose of a product of two matrices is equal to the product of their transposes in reverse order.
- In other words, (AB)` = B`A`
2. Transpose of the identity matrix: I` = I
- The transpose of the identity matrix is equal to the identity matrix itself.
Using the above properties, we can find the value of A`A:
1. Start with the given equation: AA` = I
2. Take the transpose of both sides: (AA`)` = I`
3. Apply the property of transpose of a product: (A`)`(A`) = I`
4. Apply the property of transpose of the identity matrix: A`A = I
Hence, the value of A`A is equal to the identity matrix I.
Therefore, the correct answer is option 'B'.
Given: A is a square matrix such that AA` = I.
Understanding the given information:
1. A is a square matrix: A square matrix is a matrix in which the number of rows is equal to the number of columns. Let's assume A is an n x n matrix.
2. AA` = I: Here, AA` represents the product of matrix A with its transpose A`. The result of this product is the identity matrix I.
Using properties of matrices:
1. Transpose of a product: (AB)` = B`A`
- The transpose of a product of two matrices is equal to the product of their transposes in reverse order.
- In other words, (AB)` = B`A`
2. Transpose of the identity matrix: I` = I
- The transpose of the identity matrix is equal to the identity matrix itself.
Using the above properties, we can find the value of A`A:
1. Start with the given equation: AA` = I
2. Take the transpose of both sides: (AA`)` = I`
3. Apply the property of transpose of a product: (A`)`(A`) = I`
4. Apply the property of transpose of the identity matrix: A`A = I
Hence, the value of A`A is equal to the identity matrix I.
Therefore, the correct answer is option 'B'.
= 0 are
Value of determinant 
is- a)abc
- b)0
- c)- 1
- d)a2 + bc
Correct answer is option 'B'. Can you explain this answer?
Value of determinant is
isa)
abc
b)
0
c)
- 1
d)
a2 + bc
|
Kamal Hooda answered |
How we get this answer
Then
Matrix has a value. This statement- a)is always true
- b)depends upon the order of the matrix
- c)is always false
- d)depends upon the elements of the matrix
Correct answer is option 'C'. Can you explain this answer?
Matrix has a value. This statement
a)
is always true
b)
depends upon the order of the matrix
c)
is always false
d)
depends upon the elements of the matrix
|
|
Madhavan Iyer answered |
Explanation:
The statement "Matrix has a value" is always false.
Reasoning:
A matrix is a mathematical object that consists of a rectangular array of numbers or symbols. It represents a set of values organized in rows and columns. Each individual value in a matrix is called an element.
A matrix can be empty, meaning it has no elements. In this case, the statement "Matrix has a value" would be false.
Even if a matrix is not empty, the elements within the matrix can have different values. For example, a matrix can contain all zeros, all ones, or a combination of different values. Therefore, the statement "Matrix has a value" is also false because it implies that all elements in the matrix have the same value.
Example:
Let's consider an example to illustrate this. Consider the following matrix:
1 0
0 1
This is a 2x2 matrix with four elements. Each element has a specific value: 1 or 0. Therefore, the matrix has values, but those values are not the same for all elements.
In another example, consider an empty matrix:
[]
This matrix has no elements, so it does not have any values. Therefore, the statement "Matrix has a value" is false in this case as well.
Conclusion:
In summary, the statement "Matrix has a value" is always false because a matrix can be empty or have elements with different values.
The statement "Matrix has a value" is always false.
Reasoning:
A matrix is a mathematical object that consists of a rectangular array of numbers or symbols. It represents a set of values organized in rows and columns. Each individual value in a matrix is called an element.
A matrix can be empty, meaning it has no elements. In this case, the statement "Matrix has a value" would be false.
Even if a matrix is not empty, the elements within the matrix can have different values. For example, a matrix can contain all zeros, all ones, or a combination of different values. Therefore, the statement "Matrix has a value" is also false because it implies that all elements in the matrix have the same value.
Example:
Let's consider an example to illustrate this. Consider the following matrix:
1 0
0 1
This is a 2x2 matrix with four elements. Each element has a specific value: 1 or 0. Therefore, the matrix has values, but those values are not the same for all elements.
In another example, consider an empty matrix:
[]
This matrix has no elements, so it does not have any values. Therefore, the statement "Matrix has a value" is false in this case as well.
Conclusion:
In summary, the statement "Matrix has a value" is always false because a matrix can be empty or have elements with different values.
For what value of λ, the system of equations 3x - y + z = 0, 15x - 6y + 5z = 0, x - 2y + 2z = 0 has a non-zero solution?- a)1
- b)- 2
- c)- 3
- d)None of these
Correct answer is option 'D'. Can you explain this answer?
For what value of λ, the system of equations 3x - y + z = 0, 15x - 6y + 5z = 0, x - 2y + 2z = 0 has a non-zero solution?
a)
1
b)
- 2
c)
- 3
d)
None of these
|
Shaurya Khurana answered |
Analysis:
To find the value of λ for which the given system of equations has a non-zero solution, we can rewrite the system in matrix form and then find the determinant of the coefficient matrix.
Matrix Form:
The given system of equations can be represented in matrix form as:
\[ \begin{bmatrix} 3 & -1 & 1 \\ 15 & -6 & 5 \\ 1 & -2 & 2 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \]
Finding Determinant:
To have a non-zero solution, the determinant of the coefficient matrix must be zero. Therefore, we calculate the determinant of the coefficient matrix as follows:
\[ \text{det} \begin{bmatrix} 3 & -1 & 1 \\ 15 & -6 & 5 \\ 1 & -2 & 2 \end{bmatrix} = 3(-6 \times 2 - 5 \times (-2)) - (-1)(15 \times 2 - 5 \times 1) + 1(15 \times (-2) - (-6) \times 1) \]
\[ = 3(-12 + 10) - (-15 + 5) + (30 + 6) = 3(-2) - (-10) + 36 = -6 + 10 + 36 = 40 \]
The determinant of the coefficient matrix is 40, which is non-zero. Hence, the system of equations has a non-zero solution for all values of λ.
Conclusion:
Therefore, the correct answer is option 'D' - None of these.
To find the value of λ for which the given system of equations has a non-zero solution, we can rewrite the system in matrix form and then find the determinant of the coefficient matrix.
Matrix Form:
The given system of equations can be represented in matrix form as:
\[ \begin{bmatrix} 3 & -1 & 1 \\ 15 & -6 & 5 \\ 1 & -2 & 2 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \]
Finding Determinant:
To have a non-zero solution, the determinant of the coefficient matrix must be zero. Therefore, we calculate the determinant of the coefficient matrix as follows:
\[ \text{det} \begin{bmatrix} 3 & -1 & 1 \\ 15 & -6 & 5 \\ 1 & -2 & 2 \end{bmatrix} = 3(-6 \times 2 - 5 \times (-2)) - (-1)(15 \times 2 - 5 \times 1) + 1(15 \times (-2) - (-6) \times 1) \]
\[ = 3(-12 + 10) - (-15 + 5) + (30 + 6) = 3(-2) - (-10) + 36 = -6 + 10 + 36 = 40 \]
The determinant of the coefficient matrix is 40, which is non-zero. Hence, the system of equations has a non-zero solution for all values of λ.
Conclusion:
Therefore, the correct answer is option 'D' - None of these.
The system of equations x + 2y - z = 3, 2x - 2y + 3z = 2, 3x - y + 2z = 1 has- a)no solution
- b)a unique solution
- c)an infinite number of solutions
- d)none of these
Correct answer is option 'B'. Can you explain this answer?
The system of equations x + 2y - z = 3, 2x - 2y + 3z = 2, 3x - y + 2z = 1 has
a)
no solution
b)
a unique solution
c)
an infinite number of solutions
d)
none of these
|
|
Shruti Datta answered |
Solution:
Given system of equations are:
x + 2y - z = 3 ...(1)
2x - 2y - 3z = 2 ...(2)
3x - y + 2z = 1 ...(3)
Adding equation (1) and (2), we get:
3x - 4z = 5 ...(4)
Multiplying equation (2) by 3 and adding it to equation (3), we get:
9x - 9y = 7 ...(5)
Now, multiplying equation (1) by 3 and adding it to equation (5), we get:
11x = 16
So, x = 16/11
Substituting the value of x in equation (4), we get:
z = (3x - 5)/4 = (3(16/11) - 5)/4 = 23/44
Substituting the values of x and z in equation (1), we get:
y = (3 - x + z)/2 = (3 - 16/11 + 23/44)/2 = 5/22
Therefore, the solution of the given system of equations is:
x = 16/11, y = 5/22, z = 23/44
As the solution is unique, the correct answer is option (B).
Given system of equations are:
x + 2y - z = 3 ...(1)
2x - 2y - 3z = 2 ...(2)
3x - y + 2z = 1 ...(3)
Adding equation (1) and (2), we get:
3x - 4z = 5 ...(4)
Multiplying equation (2) by 3 and adding it to equation (3), we get:
9x - 9y = 7 ...(5)
Now, multiplying equation (1) by 3 and adding it to equation (5), we get:
11x = 16
So, x = 16/11
Substituting the value of x in equation (4), we get:
z = (3x - 5)/4 = (3(16/11) - 5)/4 = 23/44
Substituting the values of x and z in equation (1), we get:
y = (3 - x + z)/2 = (3 - 16/11 + 23/44)/2 = 5/22
Therefore, the solution of the given system of equations is:
x = 16/11, y = 5/22, z = 23/44
As the solution is unique, the correct answer is option (B).
If A is a square matrix of order n such that its elements are polynomial in x and its r-rows become identical for x = k, then- a)(x - k)r is a factor of |A|
- b)(x - k)r - 1 is a factor of |A|
- c)(x - k)r + 1 is a factor of |A|
- d)(x - k)r is a factor of A
Correct answer is option 'A'. Can you explain this answer?
If A is a square matrix of order n such that its elements are polynomial in x and its r-rows become identical for x = k, then
a)
(x - k)r is a factor of |A|
b)
(x - k)r - 1 is a factor of |A|
c)
(x - k)r + 1 is a factor of |A|
d)
(x - k)r is a factor of A
|
|
Siddharth Banerjee answered |
Explanation:
Let A be a square matrix of order n such that its elements are polynomial in x. Let the r-rows of A become identical for x = k. Then we can write:
A = [a1(x), a2(x), ..., ar-1(x), ar(x), ..., an(x)]
where ar(x) = ar-1(x) for x = k. Let B be the matrix obtained from A by subtracting ar-1(x) from ar(x) in the r-th row. Then we have:
B = [a1(x), a2(x), ..., ar-1(x), 0, ..., an(x)] - [0, 0, ..., 0, ar-1(x) - ar-1(x), ..., 0]
= [a1(x), a2(x), ..., ar-1(x), 0, ..., an(x)] - [0, 0, ..., 0, 0, ..., 0]
= [a1(x), a2(x), ..., ar-1(x), ar-1(x) - ar(x), ..., an(x)]
= [a1(x), a2(x), ..., ar-1(x), (x - k)ar-1(x), ..., an(x)]
Therefore, the r-th row of B is:
[(x - k)ar-1(x), 0, ..., 0]
Now let C be the matrix obtained from B by dividing the r-th row by (x - k)r. Then we have:
C = [a1(x), a2(x), ..., ar-1(x), ar-1(x) / (x - k)r, ..., an(x)]
Therefore, the determinant of C is:
|C| = (ar-1(x) / (x - k)r) times the determinant of the matrix obtained from C by deleting the r-th row.
Since ar-1(x) = ar(x) for x = k, we have:
|C| = (ar-1(k) / (k - k)r) times the determinant of the matrix obtained from C by deleting the r-th row.
= 0 times the determinant of the matrix obtained from C by deleting the r-th row.
Therefore, (x - k)r is a factor of |C|. Since A is obtained from C by multiplying the r-th row by (x - k)r, we have:
|A| = (x - k)r times |C|
Therefore, (x - k)r is a factor of |A|. Hence, option (A) is correct.
Let A be a square matrix of order n such that its elements are polynomial in x. Let the r-rows of A become identical for x = k. Then we can write:
A = [a1(x), a2(x), ..., ar-1(x), ar(x), ..., an(x)]
where ar(x) = ar-1(x) for x = k. Let B be the matrix obtained from A by subtracting ar-1(x) from ar(x) in the r-th row. Then we have:
B = [a1(x), a2(x), ..., ar-1(x), 0, ..., an(x)] - [0, 0, ..., 0, ar-1(x) - ar-1(x), ..., 0]
= [a1(x), a2(x), ..., ar-1(x), 0, ..., an(x)] - [0, 0, ..., 0, 0, ..., 0]
= [a1(x), a2(x), ..., ar-1(x), ar-1(x) - ar(x), ..., an(x)]
= [a1(x), a2(x), ..., ar-1(x), (x - k)ar-1(x), ..., an(x)]
Therefore, the r-th row of B is:
[(x - k)ar-1(x), 0, ..., 0]
Now let C be the matrix obtained from B by dividing the r-th row by (x - k)r. Then we have:
C = [a1(x), a2(x), ..., ar-1(x), ar-1(x) / (x - k)r, ..., an(x)]
Therefore, the determinant of C is:
|C| = (ar-1(x) / (x - k)r) times the determinant of the matrix obtained from C by deleting the r-th row.
Since ar-1(x) = ar(x) for x = k, we have:
|C| = (ar-1(k) / (k - k)r) times the determinant of the matrix obtained from C by deleting the r-th row.
= 0 times the determinant of the matrix obtained from C by deleting the r-th row.
Therefore, (x - k)r is a factor of |C|. Since A is obtained from C by multiplying the r-th row by (x - k)r, we have:
|A| = (x - k)r times |C|
Therefore, (x - k)r is a factor of |A|. Hence, option (A) is correct.
unit matrixIf the system of equations 2x + 3y + 5 = 0, x + ky + 5 = 0, kx – 12y – 14 = 0 be consistent, then k =- a)– 2, 12/5
- b)– 1, 1/5
- c)– 6, 17/5
- d)6, – 12/5
Correct answer is option 'C'. Can you explain this answer?
If the system of equations 2x + 3y + 5 = 0, x + ky + 5 = 0, kx – 12y – 14 = 0 be consistent, then k =
a)
– 2, 12/5
b)
– 1, 1/5
c)
– 6, 17/5
d)
6, – 12/5
|
|
Sahana Sharma answered |
We can solve this system of equations by substitution.
From the second equation, we get x = -5/ky.
Substituting this into the first equation, we get:
2(-5/ky) + 3y + 5 = 0
Simplifying, we get:
-10/k + 3y + 5 = 0
3y = 10/k - 5
y = (10/k - 5)/3
Now, substituting x and y into the third equation, we get:
k(-5/ky) + (10/k - 5)/3 + 5 = 0
Simplifying, we get:
-5 + (10/k^2) - (5/3) + 5 = 0
10/k^2 = 5/3
k^2 = 6
k = ±√6
Therefore, the solutions are:
x = -5/(√6)y, y = (10/(√6) - 5)/3 for k = √6
x = -5/(-√6)y, y = (10/(-√6) - 5)/3 for k = -√6
From the second equation, we get x = -5/ky.
Substituting this into the first equation, we get:
2(-5/ky) + 3y + 5 = 0
Simplifying, we get:
-10/k + 3y + 5 = 0
3y = 10/k - 5
y = (10/k - 5)/3
Now, substituting x and y into the third equation, we get:
k(-5/ky) + (10/k - 5)/3 + 5 = 0
Simplifying, we get:
-5 + (10/k^2) - (5/3) + 5 = 0
10/k^2 = 5/3
k^2 = 6
k = ±√6
Therefore, the solutions are:
x = -5/(√6)y, y = (10/(√6) - 5)/3 for k = √6
x = -5/(-√6)y, y = (10/(-√6) - 5)/3 for k = -√6
If every minor of order r of a matrix A is zero, then rank of A is- a)greater than r
- b)equal to r
- c)less than or equal to r
- d)less than r
Correct answer is option 'D'. Can you explain this answer?
If every minor of order r of a matrix A is zero, then rank of A is
a)
greater than r
b)
equal to r
c)
less than or equal to r
d)
less than r
|
|
Srishti Khanna answered |
Explanation:
Rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. If every minor of order r of a matrix A is zero, it means that all submatrices of order r formed by selecting any r rows and r columns from matrix A have determinant equal to zero.
Reasoning:
- If every minor of order r of a matrix A is zero, it implies that the rows or columns of the matrix are linearly dependent.
- This means that the rank of the matrix cannot be greater than r because there cannot be more than r linearly independent rows or columns.
- However, the rank of the matrix can be less than r because even though all minors of order r are zero, it is possible that some minors of order less than r are non-zero and contribute to the rank of the matrix.
Conclusion:
Therefore, if every minor of order r of a matrix A is zero, the rank of the matrix is less than r. This is because the presence of zero minors of order r indicates linear dependence among rows or columns, limiting the maximum number of linearly independent rows or columns in the matrix to be less than r.
Rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. If every minor of order r of a matrix A is zero, it means that all submatrices of order r formed by selecting any r rows and r columns from matrix A have determinant equal to zero.
Reasoning:
- If every minor of order r of a matrix A is zero, it implies that the rows or columns of the matrix are linearly dependent.
- This means that the rank of the matrix cannot be greater than r because there cannot be more than r linearly independent rows or columns.
- However, the rank of the matrix can be less than r because even though all minors of order r are zero, it is possible that some minors of order less than r are non-zero and contribute to the rank of the matrix.
Conclusion:
Therefore, if every minor of order r of a matrix A is zero, the rank of the matrix is less than r. This is because the presence of zero minors of order r indicates linear dependence among rows or columns, limiting the maximum number of linearly independent rows or columns in the matrix to be less than r.
is
The rank of matrix 
is- a)0
- b)1
- c)2
- d)3
Correct answer is option 'C'. Can you explain this answer?
The rank of matrix is
isa)
0
b)
1
c)
2
d)
3
|
Ritika Singh answered |
Operation 1: R1<->R2
2: R2->R2-2R1
:R3->R3-5R1
:R2->R3-2R2
2: R2->R2-2R1
:R3->R3-5R1
:R2->R3-2R2
If the eigen values of a square matrix be 1, - 2 and 3, then the eigen values of the matrix 3A are- a)1, - 2, 3
- b)3, - 6, 9
- c)1/3, - 2/3, 1
- d)none of these
Correct answer is option 'B'. Can you explain this answer?
If the eigen values of a square matrix be 1, - 2 and 3, then the eigen values of the matrix 3A are
a)
1, - 2, 3
b)
3, - 6, 9
c)
1/3, - 2/3, 1
d)
none of these
|
|
Tarun Singh answered |
Eigenvalues of a matrix and scalar multiplication
Given matrix A has eigenvalues 1, -2, and 3
We need to find the eigenvalues of the matrix 3A
To find the eigenvalues of 3A, we need to use the following formula:
If λ is an eigenvalue of A, then kλ is an eigenvalue of kA, where k is a scalar.
Therefore, the eigenvalues of 3A will be:
3(1) = 3
3(-2) = -6
3(3) = 9
Hence, the eigenvalues of the matrix 3A are 3, -6, and 9, which is option B.
Given matrix A has eigenvalues 1, -2, and 3
We need to find the eigenvalues of the matrix 3A
To find the eigenvalues of 3A, we need to use the following formula:
If λ is an eigenvalue of A, then kλ is an eigenvalue of kA, where k is a scalar.
Therefore, the eigenvalues of 3A will be:
3(1) = 3
3(-2) = -6
3(3) = 9
Hence, the eigenvalues of the matrix 3A are 3, -6, and 9, which is option B.
then what is the value of sin3 A + sin3 B + sin3 C?
If A is a skew-symmetric matrix, then trace of A is- a)1
- b)- 1
- c)0
- d)none of these
Correct answer is option 'C'. Can you explain this answer?
If A is a skew-symmetric matrix, then trace of A is
a)
1
b)
- 1
c)
0
d)
none of these
|
|
Bhavana Pillai answered |
Skew-Symmetric Matrix
A skew-symmetric matrix is a square matrix where the transpose of the matrix is equal to the negative of the matrix itself. In other words, if A is a skew-symmetric matrix, then A^T = -A.
Trace of a Matrix
The trace of a matrix is defined as the sum of the elements on the main diagonal of the matrix. For an n x n matrix A, the trace is given by Tr(A) = a_11 + a_22 + ... + a_nn, where a_ij represents the element in the i-th row and j-th column of the matrix.
Proof
To prove that the trace of a skew-symmetric matrix is zero, let's consider a skew-symmetric matrix A.
A = [a_11 a_12 a_13
a_21 a_22 a_23
a_31 a_32 a_33]
Taking the transpose of A, we have:
A^T = [a_11 a_21 a_31
a_12 a_22 a_32
a_13 a_23 a_33]
According to the definition of a skew-symmetric matrix, A^T = -A. So, we can write:
-a_11 = a_11
-a_21 = a_12
-a_31 = a_13
-a_12 = a_21
-a_22 = a_22
-a_32 = a_23
-a_13 = a_31
-a_23 = a_32
-a_33 = a_33
From these equations, we can see that the diagonal elements of A must be zero (a_11 = 0, a_22 = 0, a_33 = 0).
Therefore, the trace of A, Tr(A) = a_11 + a_22 + a_33 = 0 + 0 + 0 = 0.
Conclusion
The trace of a skew-symmetric matrix is always zero. Therefore, the correct answer is option c) 0.
A skew-symmetric matrix is a square matrix where the transpose of the matrix is equal to the negative of the matrix itself. In other words, if A is a skew-symmetric matrix, then A^T = -A.
Trace of a Matrix
The trace of a matrix is defined as the sum of the elements on the main diagonal of the matrix. For an n x n matrix A, the trace is given by Tr(A) = a_11 + a_22 + ... + a_nn, where a_ij represents the element in the i-th row and j-th column of the matrix.
Proof
To prove that the trace of a skew-symmetric matrix is zero, let's consider a skew-symmetric matrix A.
A = [a_11 a_12 a_13
a_21 a_22 a_23
a_31 a_32 a_33]
Taking the transpose of A, we have:
A^T = [a_11 a_21 a_31
a_12 a_22 a_32
a_13 a_23 a_33]
According to the definition of a skew-symmetric matrix, A^T = -A. So, we can write:
-a_11 = a_11
-a_21 = a_12
-a_31 = a_13
-a_12 = a_21
-a_22 = a_22
-a_32 = a_23
-a_13 = a_31
-a_23 = a_32
-a_33 = a_33
From these equations, we can see that the diagonal elements of A must be zero (a_11 = 0, a_22 = 0, a_33 = 0).
Therefore, the trace of A, Tr(A) = a_11 + a_22 + a_33 = 0 + 0 + 0 = 0.
Conclusion
The trace of a skew-symmetric matrix is always zero. Therefore, the correct answer is option c) 0.
is
If the system of equations x - ky - z = 0, kx - y - z = 0, x + y - z = 0 has a non-zero solution, then the possible values of k are- a)- 1, 2
- b)1, 2
- c)0, 1
- d)- 1, 1
Correct answer is option 'D'. Can you explain this answer?
If the system of equations x - ky - z = 0, kx - y - z = 0, x + y - z = 0 has a non-zero solution, then the possible values of k are
a)
- 1, 2
b)
1, 2
c)
0, 1
d)
- 1, 1
|
|
Saikat Ghoshal answered |
Solution:
Given system of equations:
x - ky - z = 0
kx - y - z = 0
x + y - z = 0
Let A be the coefficient matrix of the system of equations, i.e.,
A = [1 -k -1]
[k -1 -1]
[1 1 -1]
Let B = [x]
[y]
[z]
Then, the system of equations can be written as AB = 0.
For non-zero solution, the determinant of A must be zero, i.e., det(A) = 0.
Hence, we have:
det(A) = 1(-1)(-1) + k(-1)(-1) + (1)(k)(1) - (1)(-1)(1) - k(1)(-1) - (1)(1)(-1) = 0
Simplifying the above equation, we get:
k² - 2k - 2 = 0
Solving the above quadratic equation, we get:
k = (2 ± √8)/2 = 1 ± √2
Since k must be real, we have k = 1 - √2 or k = 1 + √2.
Therefore, the possible values of k are -1, 1 + √2 and 1 - √2.
But, we know that the system of equations has non-zero solution only if k = 1, as for k = -1, the system of equations become inconsistent and for k = 1 + √2 and k = 1 - √2, the system of equations become dependent.
Hence, the only possible value of k is 1.
Therefore, the correct option is (D).
Given system of equations:
x - ky - z = 0
kx - y - z = 0
x + y - z = 0
Let A be the coefficient matrix of the system of equations, i.e.,
A = [1 -k -1]
[k -1 -1]
[1 1 -1]
Let B = [x]
[y]
[z]
Then, the system of equations can be written as AB = 0.
For non-zero solution, the determinant of A must be zero, i.e., det(A) = 0.
Hence, we have:
det(A) = 1(-1)(-1) + k(-1)(-1) + (1)(k)(1) - (1)(-1)(1) - k(1)(-1) - (1)(1)(-1) = 0
Simplifying the above equation, we get:
k² - 2k - 2 = 0
Solving the above quadratic equation, we get:
k = (2 ± √8)/2 = 1 ± √2
Since k must be real, we have k = 1 - √2 or k = 1 + √2.
Therefore, the possible values of k are -1, 1 + √2 and 1 - √2.
But, we know that the system of equations has non-zero solution only if k = 1, as for k = -1, the system of equations become inconsistent and for k = 1 + √2 and k = 1 - √2, the system of equations become dependent.
Hence, the only possible value of k is 1.
Therefore, the correct option is (D).
If a, b and c (all positive) are the pth, qth and rth terms of a geometric progression, then 
is equal to
- a)pqr
- b)0
- c)p + q + r
- d)pq + qr + rp
Correct answer is option 'B'. Can you explain this answer?
If a, b and c (all positive) are the pth, qth and rth terms of a geometric progression, then is equal to
is equal toa)
pqr
b)
0
c)
p + q + r
d)
pq + qr + rp
|
|
Veda Institute answered |
let the first term be t and common ratio be z
a=tz(p-1)
b=tz(q-1)
c=tz(r-1)
just apply these values there and apply log and matrices properties
a=tz(p-1)
b=tz(q-1)
c=tz(r-1)
just apply these values there and apply log and matrices properties
The matrix 
is invertible if- a)λ ≠ - 15
- b)λ ≠ - 17
- c)λ ≠ - 16
- d)λ ≠ - 18
Correct answer is option 'B'. Can you explain this answer?
The matrix is invertible if
is invertible ifa)
λ ≠ - 15
b)
λ ≠ - 17
c)
λ ≠ - 16
d)
λ ≠ - 18
|
Chhotu Yadav answered |
If determinant of a is non zero then the matrix is invertible.since lemda is not be -17
If a = b = c = 0, then the determinant 
is divisible by- a)x3
- b)x2
- c)(a2 + b2 + c2)
- d)none of these
Correct answer is option 'A'. Can you explain this answer?
If a = b = c = 0, then the determinant is divisible by
is divisible bya)
x3
b)
x2
c)
(a2 + b2 + c2)
d)
none of these
|
Simran Vanjani answered |
A = b = c = 0 therefore determinant would have all values equal to zero except diagonal values. diagonal values are all x. take x common from each row therefore det =| x³ * I | = x³ * 1 = x³
If A is a square matrix of order 4, and I is a unit matrix, then it is true that- a)det (2 A) = 16 det (A)
- b)det (- A) = - det (A)
- c)det (2A) = 2 det (A)
- d)det (A + 1) = det A + 1
Correct answer is option 'A'. Can you explain this answer?
If A is a square matrix of order 4, and I is a unit matrix, then it is true that
a)
det (2 A) = 16 det (A)
b)
det (- A) = - det (A)
c)
det (2A) = 2 det (A)
d)
det (A + 1) = det A + 1
|
Ayush Gupta answered |
Explanation:
To understand why option 'A' is the correct answer, let's consider the properties of the determinant.
1. Determinant of a Scalar Multiple:
The determinant of a matrix multiplied by a scalar is equal to the scalar raised to the power of the order of the matrix multiplied by the determinant of the matrix itself.
2. Determinant of a Negative Matrix:
The determinant of a matrix with all its elements negated is equal to the negative of the determinant of the original matrix.
Now, let's analyze each option:
a) det (2A) = 16 det (A)
The determinant of a matrix multiplied by a scalar is equal to the scalar raised to the power of the order of the matrix multiplied by the determinant of the matrix itself.
In this case, we have a scalar of 2 multiplying matrix A. Since the order of matrix A is 4, the determinant of 2A would be (2^4) * det(A) = 16 * det(A).
b) det (-A) = -det(A)
The determinant of a matrix with all its elements negated is equal to the negative of the determinant of the original matrix.
In this case, we have all elements of matrix A negated. So, the determinant of -A would be equal to the negative of the determinant of A, which is -det(A).
c) det (2A) = 2 det (A)
This statement is not true. The determinant of a matrix multiplied by a scalar is equal to the scalar raised to the power of the order of the matrix multiplied by the determinant of the matrix itself. In this case, the scalar is 2 and the order of matrix A is 4. So, the determinant of 2A would be (2^4) * det(A) = 16 * det(A), not 2 * det(A).
d) det (A 1) = det(A 1)
This statement is not true. The determinant of a matrix subtracted by another matrix is not equal to the determinant of the original matrix subtracted by the determinant of the other matrix.
Therefore, the correct answer is option 'A': det (2A) = 16 det (A).
To understand why option 'A' is the correct answer, let's consider the properties of the determinant.
1. Determinant of a Scalar Multiple:
The determinant of a matrix multiplied by a scalar is equal to the scalar raised to the power of the order of the matrix multiplied by the determinant of the matrix itself.
2. Determinant of a Negative Matrix:
The determinant of a matrix with all its elements negated is equal to the negative of the determinant of the original matrix.
Now, let's analyze each option:
a) det (2A) = 16 det (A)
The determinant of a matrix multiplied by a scalar is equal to the scalar raised to the power of the order of the matrix multiplied by the determinant of the matrix itself.
In this case, we have a scalar of 2 multiplying matrix A. Since the order of matrix A is 4, the determinant of 2A would be (2^4) * det(A) = 16 * det(A).
b) det (-A) = -det(A)
The determinant of a matrix with all its elements negated is equal to the negative of the determinant of the original matrix.
In this case, we have all elements of matrix A negated. So, the determinant of -A would be equal to the negative of the determinant of A, which is -det(A).
c) det (2A) = 2 det (A)
This statement is not true. The determinant of a matrix multiplied by a scalar is equal to the scalar raised to the power of the order of the matrix multiplied by the determinant of the matrix itself. In this case, the scalar is 2 and the order of matrix A is 4. So, the determinant of 2A would be (2^4) * det(A) = 16 * det(A), not 2 * det(A).
d) det (A 1) = det(A 1)
This statement is not true. The determinant of a matrix subtracted by another matrix is not equal to the determinant of the original matrix subtracted by the determinant of the other matrix.
Therefore, the correct answer is option 'A': det (2A) = 16 det (A).
If A and B are two non-zero square matrices such that AB = 0, then- a)both A and B are singular
- b)either A or B is singular
- c)neither A nor B is singular
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
If A and B are two non-zero square matrices such that AB = 0, then
a)
both A and B are singular
b)
either A or B is singular
c)
neither A nor B is singular
d)
None of these
|
Devika Choudhary answered |
Explanation:
To understand why option B is the correct answer, let's analyze the given information step by step.
Given: A and B are two non-zero square matrices such that AB = 0.
Now, let's consider the possibilities for the singularity of matrices A and B.
1. If A is non-singular:
If A is non-singular, it means that A has an inverse denoted as A^(-1). Multiplying both sides of the equation AB = 0 by A^(-1), we get:
A^(-1) * (AB) = A^(-1) * 0
B = 0
Here, we obtain B = 0, indicating that B is a zero matrix. However, the given information states that B is a non-zero matrix, which contradicts the assumption that A is non-singular. Therefore, A cannot be non-singular.
2. If B is non-singular:
If B is non-singular, it means that B has an inverse denoted as B^(-1). Multiplying both sides of the equation AB = 0 by B^(-1), we get:
A * (B * B^(-1)) = 0
A * I = 0
A = 0
Here, we obtain A = 0, indicating that A is a zero matrix. However, the given information states that A is a non-zero matrix, which contradicts the assumption that B is non-singular. Therefore, B cannot be non-singular.
From the above analysis, we can conclude that neither A nor B can be non-singular. Hence, the correct answer is option B: either A or B is singular.
Conclusion:
The correct answer is option B - either A or B is singular. This conclusion is reached by analyzing the possibilities for the singularity of matrices A and B based on the given information AB = 0.
To understand why option B is the correct answer, let's analyze the given information step by step.
Given: A and B are two non-zero square matrices such that AB = 0.
Now, let's consider the possibilities for the singularity of matrices A and B.
1. If A is non-singular:
If A is non-singular, it means that A has an inverse denoted as A^(-1). Multiplying both sides of the equation AB = 0 by A^(-1), we get:
A^(-1) * (AB) = A^(-1) * 0
B = 0
Here, we obtain B = 0, indicating that B is a zero matrix. However, the given information states that B is a non-zero matrix, which contradicts the assumption that A is non-singular. Therefore, A cannot be non-singular.
2. If B is non-singular:
If B is non-singular, it means that B has an inverse denoted as B^(-1). Multiplying both sides of the equation AB = 0 by B^(-1), we get:
A * (B * B^(-1)) = 0
A * I = 0
A = 0
Here, we obtain A = 0, indicating that A is a zero matrix. However, the given information states that A is a non-zero matrix, which contradicts the assumption that B is non-singular. Therefore, B cannot be non-singular.
From the above analysis, we can conclude that neither A nor B can be non-singular. Hence, the correct answer is option B: either A or B is singular.
Conclusion:
The correct answer is option B - either A or B is singular. This conclusion is reached by analyzing the possibilities for the singularity of matrices A and B based on the given information AB = 0.
is
= 
is equal to
where a = i, b = ω, c = ω2, then Δ is equal to
is
then 
is
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