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All questions of Linear Algebra for Civil Engineering (CE) Exam
If A is a non–singular matrix and the eigen values of A are 2 , 3 , -3 then the eigen values of A-1 are- a)2 , 3 , - 3
- b)1/2, 1/3, -1/3
- c)2|A|, 3|A|, -3|A|
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
If A is a non–singular matrix and the eigen values of A are 2 , 3 , -3 then the eigen values of A-1 are
a)
2 , 3 , - 3
b)
1/2, 1/3, -1/3
c)
2|A|, 3|A|, -3|A|
d)
None of these
|
Tanishq Chavan answered |
If λ1 ,λ2 ,λ3 ....λn are the eigen values of a non–singular matrix A, then A-1 has the eigen values 1/λ1 ,1/λ2 ,1/λ3 ....1/λn Thus eigen values of A-1are 1/2, 1/3, -1/3
Given the matrix 
the eigenvector is- a)
- b)
- c)
- d)
Correct answer is 'C'. Can you explain this answer?
Given the matrix the eigenvector is
the eigenvector isa)
b)
c)
d)
|
Ritika Bose answered |
Characteristic equation
The number of linearly independent eigenvectors of 
- a)0
- b)1
- c)2
- d)Infinit e
Correct answer is 'B'. Can you explain this answer?
The number of linearly independent eigenvectors of
a)
0
b)
1
c)
2
d)
Infinit e
|
Pranjal Sen answered |
Number of linear independent vectors is equal to the sum of Geometric Multiplicity of eigen values. Here only eigen value is 2. To find Geometric multiplicity find n-r of (matrix-2I), where n is order and r is rank. Rank of obtained matrix is 1 and n=2 so n-r=1. Therefore the no of linearly independent eigen vectors is 1
The eigenvalues of
- a)-19,5,37
- b)19,-5,-37
- c)2,-3,7
- d)3,-5,37
Correct answer is option 'A'. Can you explain this answer?
The eigenvalues of
a)
-19,5,37
b)
19,-5,-37
c)
2,-3,7
d)
3,-5,37
|
|
Avinash Sharma answered |
The eigenvalues of an upper triangular matrix are simply the diagonal entries of the matrix.
Hence 5, -19, and 37 are the eigenvalues of the matrix. Alternately, look atd
λ = 5, -19, 37
If -1 , 2 , 3 are the eigen values of a square matrix A then the eigen values of A2 are- a)-1 , 2 , 3
- b)1, 4, 9
- c)1, 2, 3
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
If -1 , 2 , 3 are the eigen values of a square matrix A then the eigen values of A2 are
a)
-1 , 2 , 3
b)
1, 4, 9
c)
1, 2, 3
d)
None of these
|
Aditya Patel answered |
If λ1 ,λ2 ,λ3 ....λn are the eigen values of a matrix A, then A2 has the eigen values λ12 ,λ22 ,λ32 ....λn2 So, eigen values of A2 are 1, 4, 9.
For a skew symmetric even ordered matrix A of integers, which of the following will not hold true:- a)det(A) = 7
- b)det(A) = 81
- c)det(A) = 9
- d)det(A) = 4
Correct answer is option 'A'. Can you explain this answer?
For a skew symmetric even ordered matrix A of integers, which of the following will not hold true:
a)
det(A) = 7
b)
det(A) = 81
c)
det(A) = 9
d)
det(A) = 4
|
Mrinalini Chavan answered |
A skew symmetric matrix Anxn is a matrix with AT = -A. The matrix of (A) satisfy this condition.
Eigen values of a matrix 
are 5 and 1. What are the eigen values of the matrix S2 = SS?- a)1 and 25
- b)6 and 4
- c)5 and 1
- d)2 and 10
Correct answer is option 'A'. Can you explain this answer?
Eigen values of a matrix are 5 and 1. What are the eigen values of the matrix S2 = SS?
are 5 and 1. What are the eigen values of the matrix S2 = SS?a)
1 and 25
b)
6 and 4
c)
5 and 1
d)
2 and 10
|
EduRev GATE answered |
We know If λ be the eigen value of A ⇒λ2 is an eigen value of A2 .
For S matrix, if eigen values are λ1, λ2, λ3,... then for S² matrix, the eigen values will be λ²1 λ²2 λ²3......
For S matrix, if eigen values are 1 and 5 then for S² matrix, the eigen values are 1 and 25
For S matrix, if eigen values are 1 and 5 then for S² matrix, the eigen values are 1 and 25
The three characteristic roots of the following matrix A 
- a)2,3
- b)1,2,2
- c)1,0,0
- d)0,2,3
Correct answer is option 'B'. Can you explain this answer?
The three characteristic roots of the following matrix A
a)
2,3
b)
1,2,2
c)
1,0,0
d)
0,2,3
|
Kabir Verma answered |
A is lower triangular matrix. So eigen values are only the diagonal elements.
the eigen value are
If 2 and 4 are the eigen values of A then the eigenvalues of AT are
- a)1/2, 1/4
- b)2, 4
- c)4, 16
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
If 2 and 4 are the eigen values of A then the eigenvalues of AT are
a)
1/2, 1/4
b)
2, 4
c)
4, 16
d)
None of these
|
Sanvi Kapoor answered |
The eigenvalues of a matrix A are the roots of the characteristic equation of A, which are invariant under the transpose operation. This means that the matrix A and its transpose AT have the same eigenvalues.
Given that 2 and 4 are the eigenvalues of A, the eigenvalues of AT will also be 2 and 4.
are –2 and 6, respectively. What is the other eigen value? 
In an M × N matrix such that all non-zero entries are covered in a rows and b column. Then the maximum number of non-zero entries, such that no two are on the same row or column, is - a)≤ a + b
- b)≤ max (a, b)
- c)≤ min[ M –a, N–b]
- d)≤ min {a, b}
Correct answer is option 'D'. Can you explain this answer?
In an M × N matrix such that all non-zero entries are covered in a rows and b column. Then the maximum number of non-zero entries, such that no two are on the same row or column, is
a)
≤ a + b
b)
≤ max (a, b)
c)
≤ min[ M –a, N–b]
d)
≤ min {a, b}
|
Naroj Boda answered |
Suppose a < b, for example let a = 3, b= 5, then we can put non-zero entries only in 3 rows and 5 columns. So suppose we put non-zero entries in any 3 rows in 3 different columns. Now we can’t put any other non-zero entry anywhere in matrix, because if we put it in some other row, then we will have 4 rows containing non-zeros, if we put it in one of those 3 rows, then we will have more than one non-zero entry in one row, which is not allowed.
So we can fill only “a” non-zero entries if a < b, similarly if b < a, we can put only “b” non-zero entries. So answer is ≤min(a,b), because whatever is less between a and b, we can put atmost that many non-zero entries.
The state variable description of a linear autonomous system is, X= AX, Where X is the two dimensional state vector and A is the system matrix given by 
The roots of the characteristic equation are - a)-2 and +2
- b)-j2 and +j2
- c)-2 and -2
- d)+2 and +2
Correct answer is option 'A'. Can you explain this answer?
The state variable description of a linear autonomous system is, X= AX,
Where X is the two dimensional state vector and A is the system matrix given by
The roots of the characteristic equation are
a)
-2 and +2
b)
-j2 and +j2
c)
-2 and -2
d)
+2 and +2
|
Janhavi Sarkar answered |
Characteristic equation will be :λ2 -4 =0 thus root of characteristic equation will be +2 and - 2.
If the rank of a (5×6) matrix Q is 4, then which one of the following statements is correct? - a)Q will have four linearly independent rows and four linearly independent columns
- b)Q will have four lineally independent rows and five lineally independent columns
- c)QQT will be invertible
- d)QTQ will be invertible
Correct answer is option 'A'. Can you explain this answer?
If the rank of a (5×6) matrix Q is 4, then which one of the following statements is correct?
a)
Q will have four linearly independent rows and four linearly independent columns
b)
Q will have four lineally independent rows and five lineally independent columns
c)
QQT will be invertible
d)
QTQ will be invertible
|
Kaavya Sengupta answered |
Given that rank Q is 4, so it will have 4 linearly independent columns as well as 4 linearly independent rows (∵ row rank of a matrix = column rank of a matrix)
So, A is correct and accordingly B is false.
C is false as order of QQ^T will be 5 x 5 and given that rank of Q is 4 i.e. < 5. So the matrix is QQ^T will be singular and hence not invertible.
Similarly D is false as order of Q^T Q will be 6 x 6 and the matrix is Q^T Q will be singular and hence not invertible.
The eigen values of the following matrix are 
- a)3, 3 + 5j, –6 – j
- b)–6 + 5j, 3 + j, 3 – j
- c)3+ j, 3 – j, 5 + j
- d)3, –1 + 3j, –1 – 3j
Correct answer is option 'D'. Can you explain this answer?
The eigen values of the following matrix are
a)
3, 3 + 5j, –6 – j
b)
–6 + 5j, 3 + j, 3 – j
c)
3+ j, 3 – j, 5 + j
d)
3, –1 + 3j, –1 – 3j
|
|
Janhavi Singh answered |
Let the matrix be A. We know, Trace (A)=sum of eigen values.
If 1 and 3 are the eigenvalues of a square matrix A then A3 is equal to- a)13 (A - I2 )
- b)13A - 12 I2
- c)12 (A - I2)
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
If 1 and 3 are the eigenvalues of a square matrix A then A3 is equal to
a)
13 (A - I2 )
b)
13A - 12 I2
c)
12 (A - I2)
d)
None of these
|
Pankaj Rane answered |
Since 1 and 3 are the eigenvalues of A so the characteristic equation of A is
Also, by Cayley–Hamilton theorem, every square matrix satisfies its own characteristic equation so
Cayley - Hamiltion Theorem states that square matrix satisfies its own characteristic equation, Consider a matrix
A satisfies the relation
- a)A +3I + 2A -2 =0
- b)A2+2A +2I=0
- c)(A+1I)( A+2I)=0
- d)exp(A) =0
Correct answer is option 'C'. Can you explain this answer?
Cayley - Hamiltion Theorem states that square matrix satisfies its own characteristic equation, Consider a matrix
A satisfies the relation
a)
A +3I + 2A -2 =0
b)
A2+2A +2I=0
c)
(A+1I)( A+2I)=0
d)
exp(A) =0
|
Telecom Tuners answered |
Characteristic equation of A is
are nonzero, and one of its eigen values is zero. Which of the following statements is true?
For which value of x will the matrix given below become singular? 
- a)4
- b)6
- c)8
- d)12
Correct answer is option 'A'. Can you explain this answer?
For which value of x will the matrix given below become singular?
a)
4
b)
6
c)
8
d)
12
|
Gate Funda answered |
Let the given matrix be A. A is singular.
The sum of the eigenvalues of 
is equal to - a)18
- b)15
- c)10
- d)none of these
Correct answer is option 'A'. Can you explain this answer?
The sum of the eigenvalues of is equal to
is equal to a)
18
b)
15
c)
10
d)
none of these
|
Zoya Sharma answered |
- Since the sum of the eigenvalues of an n–square matrix is equal to the trace of the matrix (i.e. sum of the diagonal elements)
- So, required sum = 8 + 5 + 5 = 18
Solution for the system defined by the set of equations
4y + 3z = 8;
2x – z = 2
and 3x + 2y =5 is - a)x = 0; y = 1; z = 4/3
- b)x = 0; y = 1/2 ; z = 2
- c)x = 1; y = 1/2 ; z = 2
- d)non-existent
Correct answer is option 'D'. Can you explain this answer?
Solution for the system defined by the set of equations
4y + 3z = 8;
2x – z = 2
and 3x + 2y =5 is
4y + 3z = 8;
2x – z = 2
and 3x + 2y =5 is
a)
x = 0; y = 1; z = 4/3
b)
x = 0; y = 1/2 ; z = 2
c)
x = 1; y = 1/2 ; z = 2
d)
non-existent
|
Dishani Desai answered |
Ans.(d)
Consider the matrix A =
,Now det (A) = 0
So byCramer's Rule the system has no solution
Consider the matrix A =
,Now det (A) = 0So byCramer's Rule the system has no solution
The following system of equations 
has a unique solution. The only possible value(s) for a is/are - a)0
- b)either 0 or 1
- c)one of 0, 1 or –1
- d)any real number other than 5
Correct answer is option 'D'. Can you explain this answer?
The following system of equations
has a unique solution. The only possible value(s) for a is/are
a)
0
b)
either 0 or 1
c)
one of 0, 1 or –1
d)
any real number other than 5
|
|
Gate Funda answered |
System has unique Soln if rank (A) = rank ( A ) = 3 . It is possible if a ≠ 5.
the eigen value corresponding to the eigenvector 
The number of linearly independent eigenvectors of - a)0
- b)1
- c)2
- d)Infinit e
Correct answer is option 'B'. Can you explain this answer?
The number of linearly independent eigenvectors of
a)
0
b)
1
c)
2
d)
Infinit e
|
|
Ravi Singh answered |
Number of linear independent vectors is equal to the sum of Geometric Multiplicity of eigen values. Here only eigen value is 2. To find Geometric multiplicity find n-r of (matrix-2I), where n is order and r is rank. Rank of obtained matrix is 1 and n=2 so n-r=1. Therefore the no of linearly independent eigen vectors is 1
then the value of x is
Find the Eigenvalues of matrix
- a)8, 2
- b)3, -3
- c)-3, -5
- d)5, 0
Correct answer is option 'A'. Can you explain this answer?
Find the Eigenvalues of matrix
a)
8, 2
b)
3, -3
c)
-3, -5
d)
5, 0
|
|
Sanya Agarwal answered |
A - λI = 0
Now, After taking the determinant:
(4 - λ)2 - 1 = 0
16 + λ2 - 8λ - 1 = 0
λ2 - 8λ + 15 = 0
(λ
- 3) (
λ - 5) = 0
λ
= 3, 5
s one of the eigen values is equal to -2. Which of the following is an eigen vector? 
The eigen values and the corresponding eigen vectors of a 2 × 2 matrix are given by
- a)
- b)
- c)
- d)
Correct answer is option 'A'. Can you explain this answer?
The eigen values and the corresponding eigen vectors of a 2 × 2 matrix are given by
a)
b)
c)
d)
|
Subodh Sharma answered |
Trace = sum of diagonal elements = sum of eigen value = 12
Only option (a) is satisfied.
Only option (a) is satisfied.
Find the rank of the matrix
- a)2
- b)1
- c)0
- d)3
Correct answer is option 'D'. Can you explain this answer?
Find the rank of the matrix
a)
2
b)
1
c)
0
d)
3
|
Engineers Adda answered |
To find out the rank of the matrix first find the |A|
If the value of the |A| = 0 then the matrix is said to be reduced
But, as the determinant of A has some finite value, then the rank of the matrix is 3.
If the value of the |A| = 0 then the matrix is said to be reduced
But, as the determinant of A has some finite value, then the rank of the matrix is 3.
then top row of R-1 is 
The system of linear equations
4x + 2y = 7
2x + y = 6 has
- a)A unique solution
- b)no solution
- c)An infinite number of solutions
- d)exactly two distinct solutions
Correct answer is option 'B'. Can you explain this answer?
The system of linear equations
4x + 2y = 7
2x + y = 6 has
4x + 2y = 7
2x + y = 6 has
a)
A unique solution
b)
no solution
c)
An infinite number of solutions
d)
exactly two distinct solutions
|
|
Rhea Reddy answered |
(b) This can be written as AX = B Where A
Angemented matrix
rank(A) ≠ rank(
). The system is inconsistant .So system has no solution.
). The system is inconsistant .So system has no solution.
For the following set of simultaneous equations:
1.5x – 0.5y = 2
4x + 2y + 3z = 9
7x + y + 5z = 10 - a)The solution is unique
- b)Infinitely many solutions exist
- c)The equations are incompatible
- d)Finite number of multiple solutions exist
Correct answer is option 'A'. Can you explain this answer?
For the following set of simultaneous equations:
1.5x – 0.5y = 2
4x + 2y + 3z = 9
7x + y + 5z = 10
1.5x – 0.5y = 2
4x + 2y + 3z = 9
7x + y + 5z = 10
a)
The solution is unique
b)
Infinitely many solutions exist
c)
The equations are incompatible
d)
Finite number of multiple solutions exist
|
Dipika Nambiar answered |
(a)
∴ rank of(
) = rank of(A) = 3
) = rank of(A) = 3∴ The system has unique solution.
Select a suitable figure from the four alternatives that would complete the figure matrix.
- a) 1
- b) 2
- c) 3
- d)4
Correct answer is option 'D'. Can you explain this answer?
Select a suitable figure from the four alternatives that would complete the figure matrix.
a)
1
b)
2
c)
3
d)
4
|
|
Kiran Datta answered |
In each row (as well as each column), the third figure is a combination of all the, elements of the first and the second figures
is,
The eigenvalues of
are- a)3, -5, 37
- b)2, -3, 7
- c)19, -5, -37
- d)-19, 5, 37
Correct answer is option 'D'. Can you explain this answer?
The eigenvalues of
are
a)
3, -5, 37
b)
2, -3, 7
c)
19, -5, -37
d)
-19, 5, 37
|
|
Avinash Sharma answered |
The eigenvalues of an upper triangular matrix are simply the diagonal entries of the matrix. Hence 5, -19, and 37 are the eigenvalues of the matrix. Alternately, look at 
Then 
= 5,-19,37 are the roots of the equation; and hence, the eigenvalues of [A].
= 5,-19,37 are the roots of the equation; and hence, the eigenvalues of [A].
The matrix A
= 
is decomposed into a product of a lower triangular matrix [L] and an upper triangular matrix [U]. The properly decomposed [L] and [U] matrices respectively are
- a)
- b)
- c)
- d)
Correct answer is option 'D'. Can you explain this answer?
The matrix A
=
is decomposed into a product of a lower triangular matrix [L] and an upper triangular matrix [U]. The properly decomposed [L] and [U] matrices respectively are
=
is decomposed into a product of a lower triangular matrix [L] and an upper triangular matrix [U]. The properly decomposed [L] and [U] matrices respectively are
a)
b)
c)
d)
|
Byomkesh Kamila answered |
As,in option d, the product of the given matrice yields the matrix A.
For what value of λ, do the simultaneous equation 2x + 3y = 1, 4x + 6y = λ have infinite solutions?- a)λ = 0
- b)λ = 1
- c)λ ≠ 2
- d)λ = 2
Correct answer is option 'D'. Can you explain this answer?
For what value of λ, do the simultaneous equation 2x + 3y = 1, 4x + 6y = λ have infinite solutions?
a)
λ = 0
b)
λ = 1
c)
λ ≠ 2
d)
λ = 2
|
Sanskriti Datta answered |
Understanding Infinite Solutions
To determine the value of λ for which the simultaneous equations have infinite solutions, we need to analyze the equations:
1. Original Equations:
- 2x + 3y = 1 (Equation 1)
- 4x + 6y = λ (Equation 2)
2. Condition for Infinite Solutions:
- For two linear equations to have infinite solutions, they must be equivalent. This means that one equation can be expressed as a multiple of the other.
Transforming the Equations
- We can manipulate Equation 1 to see if it can transform into Equation 2:
- Multiply Equation 1 by 2:
- 2 * (2x + 3y) = 2 * 1
- 4x + 6y = 2
Setting Equations Equal
- Now we compare this with Equation 2:
- From our manipulation, we get:
- 4x + 6y = 2
- For the two equations to be equivalent:
- λ must equal 2.
Conclusion
- Therefore, the value of λ that allows the equations to have infinite solutions is:
- λ = 2.
Thus, the correct option is D) λ = 2.
To determine the value of λ for which the simultaneous equations have infinite solutions, we need to analyze the equations:
1. Original Equations:
- 2x + 3y = 1 (Equation 1)
- 4x + 6y = λ (Equation 2)
2. Condition for Infinite Solutions:
- For two linear equations to have infinite solutions, they must be equivalent. This means that one equation can be expressed as a multiple of the other.
Transforming the Equations
- We can manipulate Equation 1 to see if it can transform into Equation 2:
- Multiply Equation 1 by 2:
- 2 * (2x + 3y) = 2 * 1
- 4x + 6y = 2
Setting Equations Equal
- Now we compare this with Equation 2:
- From our manipulation, we get:
- 4x + 6y = 2
- For the two equations to be equivalent:
- λ must equal 2.
Conclusion
- Therefore, the value of λ that allows the equations to have infinite solutions is:
- λ = 2.
Thus, the correct option is D) λ = 2.
There are two containers, with one containing 4 Red and 3 Green balls and the other containing 3 Blue and 4 Green balls. One bal is drawn at random form each container.The probability that one of the ball is Red and the other is Blue will be
- a)1/7
- b)9/49
- c)12/49
- d)3/7
Correct answer is option 'C'. Can you explain this answer?
There are two containers, with one containing 4 Red and 3 Green balls and the other containing 3 Blue and 4 Green balls. One bal is drawn at random form each container.The probability that one of the ball is Red and the other is Blue will be
a)
1/7
b)
9/49
c)
12/49
d)
3/7
|
|
Snehal Nair answered |
Problem Analysis
To solve this problem, we need to find the probability of drawing one Red ball and one Blue ball from two different containers. Let's analyze the given information:
- Container 1: 4 Red balls and 3 Green balls
- Container 2: 3 Blue balls and 4 Green balls
Solution
Step 1: Find the probability of drawing a Red ball from Container 1
The probability of drawing a Red ball from Container 1 can be calculated as the ratio of the number of Red balls to the total number of balls in Container 1.
Probability of drawing a Red ball from Container 1 = 4/7
Step 2: Find the probability of drawing a Blue ball from Container 2
Similarly, the probability of drawing a Blue ball from Container 2 can be calculated as the ratio of the number of Blue balls to the total number of balls in Container 2.
Probability of drawing a Blue ball from Container 2 = 3/7
Step 3: Find the probability of drawing one Red and one Blue ball
Now, we need to find the probability of drawing one Red ball from Container 1 and one Blue ball from Container 2. Since the two events are independent, we can multiply the probabilities of each event.
Probability of drawing one Red and one Blue ball = Probability of drawing a Red ball from Container 1 × Probability of drawing a Blue ball from Container 2
Probability of drawing one Red and one Blue ball = (4/7) × (3/7) = 12/49
Therefore, the probability that one of the balls is Red and the other is Blue is 12/49.
Conclusion
The correct answer is option 'C', 12/49.
The system of equationsx + y + z = 6;x + 4y + 6z = 20;x + 4y + λz = μhas NO solution for values of λ and μ given by- a)λ = 6, μ ≠ 20
- b)λ = 6, μ = 20
- c)λ ≠ 6, μ = 20
- d)λ ≠ 6, μ ≠ 20
Correct answer is option 'A'. Can you explain this answer?
The system of equations
x + y + z = 6;
x + 4y + 6z = 20;
x + 4y + λz = μ
has NO solution for values of λ and μ given by
a)
λ = 6, μ ≠ 20
b)
λ = 6, μ = 20
c)
λ ≠ 6, μ = 20
d)
λ ≠ 6, μ ≠ 20
|
Meghana Kaur answered |
Concept:
The number of solutions can be determined by finding out the rank of the Augmented matrix and the rank of the Coefficient matrix.
- If rank(Augmented matrix) = rank(Coefficient matrix) = no. of variables then no of solutions = 1.
- If rank(Augmented matrix) ≠ rank(Coefficient matrix) then no of solutions = 0.
- If rank(Augmented matrix) = rank(Coefficient matrix) < no. of variables, no of solutions = infinite.
Calculation:
The augmented matrix for the system of equations is
Performing: R3 → R3 – R2
…If λ = 6 and μ ≠ 20 then
Rank (A | B) = 3 and Rank (A) = 2
∵ Rank (A | B) ≠ Rank (A)
∴ Given the system of equations has no solution for λ = 6 and μ ≠ 20
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