All Exams >
Mathematics >
Topic-wise Tests & Solved Examples for Mathematics >
All Questions
All questions of Matrix for Mathematics Exam
Square matrix A of order n over R has rank n. Which one of the following statement is not correct?- a)AT has rank n
- b)A has n linearly independent columns
- c)A is non — singular
- d)A is singular
Correct answer is option 'D'. Can you explain this answer?
Square matrix A of order n over R has rank n. Which one of the following statement is not correct?
a)
AT has rank n
b)
A has n linearly independent columns
c)
A is non — singular
d)
A is singular
|
Chirag Verma answered |
Since, if A is a square matrix of rank n, then it cannot be a singular
is 
A square matrix A such that AT = -A, is called a- a)Symmetric matrix
- b)Hermitian Matrix
- c)Skew Hermitian Matrix
- d)Skew Symmetric matrix
Correct answer is option 'D'. Can you explain this answer?
A square matrix A such that AT = -A, is called a
a)
Symmetric matrix
b)
Hermitian Matrix
c)
Skew Hermitian Matrix
d)
Skew Symmetric matrix
|
|
Chirag Verma answered |
In mathematics, particularly in linear algebra, a skew-symmetric matrix is a square matrix whose transpose equals its negative.
A matrix is- a)A real number
- b)Either a real or a complex number
- c)An arrangement of mn numbers, over the given set S, in a rectangular array of m rows and n Columns.
- d)Same as determinant
Correct answer is option 'C'. Can you explain this answer?
A matrix is
a)
A real number
b)
Either a real or a complex number
c)
An arrangement of mn numbers, over the given set S, in a rectangular array of m rows and n Columns.
d)
Same as determinant
|
Urvi Shah answered |
Explanation:
Definition of a Matrix:
- A matrix is an arrangement of mn numbers, over a given set S, in a rectangular array of m rows and n columns.
Characteristics of a Matrix:
- Each individual number in a matrix is called an element or entry.
- The numbers in a matrix can be real numbers, complex numbers, or elements from any other set.
- Matrices are widely used in various fields such as mathematics, physics, computer science, and engineering.
Types of Matrices:
- There are different types of matrices such as square matrices, row matrices, column matrices, diagonal matrices, etc.
- The properties and operations of matrices depend on the type of matrix.
Matrix Operations:
- Addition, subtraction, multiplication, and division are some of the key operations performed on matrices.
- Matrix multiplication is different from scalar multiplication and involves multiplying corresponding elements and summing them up.
Application of Matrices:
- Matrices are used in solving systems of linear equations, representing transformations in geometry, analyzing networks, and many other applications.
- They provide a compact and efficient way to represent and manipulate data in various mathematical and computational problems.
In conclusion, a matrix is an arrangement of numbers in a rectangular array, and it plays a crucial role in various mathematical and computational applications.
Definition of a Matrix:
- A matrix is an arrangement of mn numbers, over a given set S, in a rectangular array of m rows and n columns.
Characteristics of a Matrix:
- Each individual number in a matrix is called an element or entry.
- The numbers in a matrix can be real numbers, complex numbers, or elements from any other set.
- Matrices are widely used in various fields such as mathematics, physics, computer science, and engineering.
Types of Matrices:
- There are different types of matrices such as square matrices, row matrices, column matrices, diagonal matrices, etc.
- The properties and operations of matrices depend on the type of matrix.
Matrix Operations:
- Addition, subtraction, multiplication, and division are some of the key operations performed on matrices.
- Matrix multiplication is different from scalar multiplication and involves multiplying corresponding elements and summing them up.
Application of Matrices:
- Matrices are used in solving systems of linear equations, representing transformations in geometry, analyzing networks, and many other applications.
- They provide a compact and efficient way to represent and manipulate data in various mathematical and computational problems.
In conclusion, a matrix is an arrangement of numbers in a rectangular array, and it plays a crucial role in various mathematical and computational applications.
Which of the following statements is correct?
- a)(A + B)T = AT + BT
- b)(AB)T = ATB
- c)(KA)T = KTAT
- d)none of the above
Correct answer is option 'A'. Can you explain this answer?
Which of the following statements is correct?
a)
(A + B)T = AT + BT
b)
(AB)T = ATB
c)
(KA)T = KTAT
d)
none of the above
|
Bhavya Reddy answered |
Statement: (A B)T = AT BT
To determine whether this statement is correct or not, let's break it down and analyze each part separately.
Part 1: (A B)T
- (A B) represents the product of two matrices A and B.
- The order of the product matrix (A B) is determined by the number of rows of A and the number of columns of B.
- The transpose operation (T) swaps the rows and columns of a matrix.
Part 2: AT BT
- AT represents the transpose of matrix A.
- BT represents the transpose of matrix B.
Analysis:
To prove the correctness of the statement, we need to show that (A B)T is equal to AT BT.
Let's consider the dimensions of the matrices involved:
- A has dimensions m x n.
- B has dimensions n x p.
The product matrix (A B) will have dimensions m x p.
- The transpose of (A B) will have dimensions p x m.
The transpose of A (AT) will have dimensions n x m.
The transpose of B (BT) will have dimensions p x n.
Since the number of columns of AT is equal to the number of rows of BT, we can perform the matrix multiplication.
Proof:
Let's calculate (A B)T:
- (A B) = C (let's assume C is the product matrix)
- C has dimensions m x p
To calculate (A B)T, we need to swap the rows and columns of C:
- (A B)T = CT
Since C has dimensions m x p, CT will have dimensions p x m.
Now, let's calculate AT BT:
- AT has dimensions n x m
- BT has dimensions p x n
To calculate AT BT, we need to perform the matrix multiplication:
- (AT BT) = D (let's assume D is the product matrix)
- D has dimensions p x m
Conclusion:
From the analysis above, we can see that (A B)T and AT BT have the same dimensions, p x m. Therefore, the statement (A B)T = AT BT is correct.
To determine whether this statement is correct or not, let's break it down and analyze each part separately.
Part 1: (A B)T
- (A B) represents the product of two matrices A and B.
- The order of the product matrix (A B) is determined by the number of rows of A and the number of columns of B.
- The transpose operation (T) swaps the rows and columns of a matrix.
Part 2: AT BT
- AT represents the transpose of matrix A.
- BT represents the transpose of matrix B.
Analysis:
To prove the correctness of the statement, we need to show that (A B)T is equal to AT BT.
Let's consider the dimensions of the matrices involved:
- A has dimensions m x n.
- B has dimensions n x p.
The product matrix (A B) will have dimensions m x p.
- The transpose of (A B) will have dimensions p x m.
The transpose of A (AT) will have dimensions n x m.
The transpose of B (BT) will have dimensions p x n.
Since the number of columns of AT is equal to the number of rows of BT, we can perform the matrix multiplication.
Proof:
Let's calculate (A B)T:
- (A B) = C (let's assume C is the product matrix)
- C has dimensions m x p
To calculate (A B)T, we need to swap the rows and columns of C:
- (A B)T = CT
Since C has dimensions m x p, CT will have dimensions p x m.
Now, let's calculate AT BT:
- AT has dimensions n x m
- BT has dimensions p x n
To calculate AT BT, we need to perform the matrix multiplication:
- (AT BT) = D (let's assume D is the product matrix)
- D has dimensions p x m
Conclusion:
From the analysis above, we can see that (A B)T and AT BT have the same dimensions, p x m. Therefore, the statement (A B)T = AT BT is correct.
[A] is a square matrix which is neither symmetric nor skew-symmetric and [A]T is its transpose. The sum and difference of these matrices are defined as [S] = [A] + [A]T and [D] = [A] -[A]T, respectively. Which of the following statements is True?- a)both [S] and [D] are symmetric
- b)both [S] and [D] are skew-symmetric
- c)[S] is skew-symmetric and [D] is symmetric
- d)[S] is symmetric and [D] is skew-symmetric.
Correct answer is option 'D'. Can you explain this answer?
[A] is a square matrix which is neither symmetric nor skew-symmetric and [A]T is its transpose. The sum and difference of these matrices are defined as [S] = [A] + [A]T and [D] = [A] -[A]T, respectively. Which of the following statements is True?
a)
both [S] and [D] are symmetric
b)
both [S] and [D] are skew-symmetric
c)
[S] is skew-symmetric and [D] is symmetric
d)
[S] is symmetric and [D] is skew-symmetric.
|
Atharv Inamdar answered |
is, 
is equal to 
A is a 3 x 4 real matrix and A x = b is an inconsistent system of equations. The highest possible rank of A is- a)1
- b)2
- c)3
- d)4
Correct answer is option 'B'. Can you explain this answer?
A is a 3 x 4 real matrix and A x = b is an inconsistent system of equations. The highest possible rank of A is
a)
1
b)
2
c)
3
d)
4
|
|
Chirag Verma answered |
Highest possible rank of A= 2 ,as Ax = b is an inconsistent system.
To convert a Hermitian Matrix into Skew Hermitian Matrix, the Hermitian Matrix must be multiplied by- a)-1
- b)i
- c)-i
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
To convert a Hermitian Matrix into Skew Hermitian Matrix, the Hermitian Matrix must be multiplied by
a)
-1
b)
i
c)
-i
d)
None of these
|
Falak Bhatia answered |
Conversion of a Hermitian Matrix into a Skew Hermitian Matrix
A Hermitian matrix is a square matrix that is equal to its own conjugate transpose. In other words, the matrix is equal to the complex conjugate of its own transpose. A Skew Hermitian matrix, on the other hand, is a square matrix whose conjugate transpose is equal to the negative of the matrix itself.
To convert a Hermitian matrix into a Skew Hermitian matrix, we can use the following steps:
Step 1: Take the conjugate transpose of the Hermitian matrix
- The conjugate transpose of a matrix is obtained by taking the complex conjugate of each element and then transposing the matrix.
- Let's assume our Hermitian matrix is A. The conjugate transpose of A is denoted as A*.
- Mathematically, A* = (A)^H, where (A)^H represents the complex conjugate of A.
Step 2: Multiply the conjugate transpose by 'i'
- In order to convert the Hermitian matrix into a Skew Hermitian matrix, we need to multiply the conjugate transpose by 'i', which is the imaginary unit.
- Mathematically, B = i * A*, where B is the resulting Skew Hermitian matrix.
Step 3: Verify if B is a Skew Hermitian matrix
- To verify if B is a Skew Hermitian matrix, we need to check if the conjugate transpose of B is equal to the negative of B.
- If (B)^H = -B, then B is a Skew Hermitian matrix.
Explanation of the Correct Answer
The correct answer to the given question is option 'B' (i) because when we multiply the conjugate transpose of a Hermitian matrix by 'i', we obtain a Skew Hermitian matrix.
- Option 'A' (-1) is incorrect because multiplying the Hermitian matrix by -1 would not result in a Skew Hermitian matrix.
- Option 'C' (i-d) is incorrect because subtracting a complex number from 'i' would not result in a Skew Hermitian matrix.
- Option 'D' (None of these) is incorrect because the correct answer is option 'B' (i), as explained above.
Therefore, to convert a Hermitian matrix into a Skew Hermitian matrix, the Hermitian matrix must be multiplied by 'i'.
A Hermitian matrix is a square matrix that is equal to its own conjugate transpose. In other words, the matrix is equal to the complex conjugate of its own transpose. A Skew Hermitian matrix, on the other hand, is a square matrix whose conjugate transpose is equal to the negative of the matrix itself.
To convert a Hermitian matrix into a Skew Hermitian matrix, we can use the following steps:
Step 1: Take the conjugate transpose of the Hermitian matrix
- The conjugate transpose of a matrix is obtained by taking the complex conjugate of each element and then transposing the matrix.
- Let's assume our Hermitian matrix is A. The conjugate transpose of A is denoted as A*.
- Mathematically, A* = (A)^H, where (A)^H represents the complex conjugate of A.
Step 2: Multiply the conjugate transpose by 'i'
- In order to convert the Hermitian matrix into a Skew Hermitian matrix, we need to multiply the conjugate transpose by 'i', which is the imaginary unit.
- Mathematically, B = i * A*, where B is the resulting Skew Hermitian matrix.
Step 3: Verify if B is a Skew Hermitian matrix
- To verify if B is a Skew Hermitian matrix, we need to check if the conjugate transpose of B is equal to the negative of B.
- If (B)^H = -B, then B is a Skew Hermitian matrix.
Explanation of the Correct Answer
The correct answer to the given question is option 'B' (i) because when we multiply the conjugate transpose of a Hermitian matrix by 'i', we obtain a Skew Hermitian matrix.
- Option 'A' (-1) is incorrect because multiplying the Hermitian matrix by -1 would not result in a Skew Hermitian matrix.
- Option 'C' (i-d) is incorrect because subtracting a complex number from 'i' would not result in a Skew Hermitian matrix.
- Option 'D' (None of these) is incorrect because the correct answer is option 'B' (i), as explained above.
Therefore, to convert a Hermitian matrix into a Skew Hermitian matrix, the Hermitian matrix must be multiplied by 'i'.
Two matrics Aand B are said to anti — commute if
- a)AB = BA
- b)AB2 = A2B
- c)AB = -BA
- d)A/B = B/A
Correct answer is option 'C'. Can you explain this answer?
Two matrics Aand B are said to anti — commute if
a)
AB = BA
b)
AB2 = A2B
c)
AB = -BA
d)
A/B = B/A
|
Zara Khan answered |
Two matrices A and B are said to be anti-symmetric if the transpose of matrix A is equal to the negative of matrix A, and the transpose of matrix B is equal to the negative of matrix B.
In mathematical notation, this can be expressed as:
A^T = -A
B^T = -B
In mathematical notation, this can be expressed as:
A^T = -A
B^T = -B
the rank of the matrix is
The eigen values of a skew-symmetric matrix are- a)Always zero
- b)always pure imaginary
- c)Either zero or pure imaginary
- d)always real
Correct answer is option 'C'. Can you explain this answer?
The eigen values of a skew-symmetric matrix are
a)
Always zero
b)
always pure imaginary
c)
Either zero or pure imaginary
d)
always real
|
|
Chirag Verma answered |
ANSWER :- c
Solution :- Let A be real skew symmetric and suppose λ∈C is an eigenvalue, with (complex) eigenvector v . Then, denoting by H hermitian transposition,
λvHv=vH(λv)=vH(Av)=vH(−AHv)=−(vHAH)v=−(Av)Hv=−(λv)Hv=−λ¯vHv
Since vHv≠0 , as v≠0 , we get
λ=−λ¯
so λ is purely imaginary (which includes 0). Note that the proof works the same for a antihermitian (complex) matrix.
With a completely similar technique you can prove that the eigenvalues of a Hermitian matrix (which includes real symmetric) are real.
If A is any square matrix, then A — A* is a- a)Symmetric matrix
- b)Skew symmetric matrix
- c)Hermitian Matrix
- d)Skew Hermitian Matrix
Correct answer is option 'D'. Can you explain this answer?
If A is any square matrix, then A — A* is a
a)
Symmetric matrix
b)
Skew symmetric matrix
c)
Hermitian Matrix
d)
Skew Hermitian Matrix
|
Veda Institute answered |
(A − A')' = A' − (A')'
= A' − A
= −(A − A')
Therefore, it is a skew symmetric matrix
Therefore, it is a skew symmetric matrix
The product of two Unitary matrices is a ________ matrix- a)Unit
- b)Orthogonal
- c)Unitary
- d)None of the above
Correct answer is option 'C'. Can you explain this answer?
The product of two Unitary matrices is a ________ matrix
a)
Unit
b)
Orthogonal
c)
Unitary
d)
None of the above
|
Aditi Singh answered |
Unitary Matrices Product
Unitary matrices are matrices that satisfy the property: U*U^H = I, where U^H is the conjugate transpose of U and I is the identity matrix.
Product of Two Unitary Matrices
When you multiply two unitary matrices together, the result is also a unitary matrix. This can be proven as follows:
Let A and B be two unitary matrices. The product of A and B is AB.
Taking the conjugate transpose of AB, we get (AB)^H = B^H * A^H.
Now, to show that AB is unitary, we need to check if (AB)(AB)^H = I.
(AB)(AB)^H = AB * (B^H * A^H) = A(BB^H)A^H.
Since A and B are unitary matrices, A*A^H = I and B*B^H = I.
Therefore, A(BB^H)A^H = AIA^H = AA^H = I.
Hence, the product of two unitary matrices is also a unitary matrix.
Two matrices A and B are said to be comparable if- a)Number of rows in A is equal to the number of columns in B
- b)Number of columns in A is equal to the number of rows in B
- c)Number of rows and columns in A is equal to the number of rows and columns in B
- d)None of the above
Correct answer is option 'C'. Can you explain this answer?
Two matrices A and B are said to be comparable if
a)
Number of rows in A is equal to the number of columns in B
b)
Number of columns in A is equal to the number of rows in B
c)
Number of rows and columns in A is equal to the number of rows and columns in B
d)
None of the above
|
Esha Verma answered |
Explanation:
To determine if two matrices A and B are comparable, we need to consider the dimensions of both matrices.
Matrix A:
- Let's assume that matrix A has m rows and n columns.
- The dimensions of matrix A can be written as m x n.
Matrix B:
- Let's assume that matrix B has p rows and q columns.
- The dimensions of matrix B can be written as p x q.
In order for two matrices to be comparable, the number of columns in matrix A should be equal to the number of rows in matrix B.
Therefore, the correct condition for comparability is:
Condition:
- Number of columns in A is equal to the number of rows in B (n = p).
Explanation of Answer Options:
a) Number of rows in A is equal to the number of columns in B: This condition is not correct for comparability as it is the opposite of the correct condition.
b) Number of columns in A is equal to the number of rows in B: This is the correct condition for comparability.
c) Number of rows and columns in A is equal to the number of rows and columns in B: This condition is not correct for comparability as it considers the dimensions of both matrices to be exactly the same, which is not necessary for comparability.
d) None of the above: This option is incorrect as option (b) is the correct condition for comparability.
Conclusion:
In order for two matrices A and B to be comparable, the number of columns in matrix A should be equal to the number of rows in matrix B.
To determine if two matrices A and B are comparable, we need to consider the dimensions of both matrices.
Matrix A:
- Let's assume that matrix A has m rows and n columns.
- The dimensions of matrix A can be written as m x n.
Matrix B:
- Let's assume that matrix B has p rows and q columns.
- The dimensions of matrix B can be written as p x q.
In order for two matrices to be comparable, the number of columns in matrix A should be equal to the number of rows in matrix B.
Therefore, the correct condition for comparability is:
Condition:
- Number of columns in A is equal to the number of rows in B (n = p).
Explanation of Answer Options:
a) Number of rows in A is equal to the number of columns in B: This condition is not correct for comparability as it is the opposite of the correct condition.
b) Number of columns in A is equal to the number of rows in B: This is the correct condition for comparability.
c) Number of rows and columns in A is equal to the number of rows and columns in B: This condition is not correct for comparability as it considers the dimensions of both matrices to be exactly the same, which is not necessary for comparability.
d) None of the above: This option is incorrect as option (b) is the correct condition for comparability.
Conclusion:
In order for two matrices A and B to be comparable, the number of columns in matrix A should be equal to the number of rows in matrix B.
Given an orthogonal matrix A = 
[AAT]-1 is
- a)
- b)
- c)
- d)
Correct answer is option 'C'. Can you explain this answer?
Given an orthogonal matrix A = [AAT]-1 is
[AAT]-1 is a)
b)
c)
d)
|
|
Veda Institute answered |
We know
AAt = I4
[AAT]-1 = [I4]-1 = I4
AAt = I4
[AAT]-1 = [I4]-1 = I4
Which of the following statements is or are correct. The trace of a matrix A =
is defined by Tr(A) = a11 + a22 a)Trace of a unit matrix is always a fixed natural numberb)Trace of the sum of two matrices is equal to sum of their tracesc)Trace of the scalar matrix is some multiple of its orderd)Tr(AB) = Tr(BA)Correct answer is option 'A,B,D'. Can you explain this answer?
is defined by Tr(A) = a11 + a22
|
Ayushi Yadav Ayushi Yada answered |
C
If the system of equations x — 2y — 3z = 1, ( P + 2 )z = 3, (2 P + l ) y + z = 2 inconsistent, then value of P is
- a)-2
- b)-1/2
- c)0
- d)2
Correct answer is option 'A'. Can you explain this answer?
If the system of equations x — 2y — 3z = 1, ( P + 2 )z = 3, (2 P + l ) y + z = 2 inconsistent, then value of P is
a)
-2
b)
-1/2
c)
0
d)
2
|
Aakash Singh answered |
It seems like the sentence is incomplete. Could you please provide the complete question or equations?
Eigen values of a matrix S = 
are 5 and 1. What is 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 S = are 5 and 1. What is the eigen values of the matrix S2 = SS?
are 5 and 1. What is the eigen values of the matrix S2 = SS?a)
1 and 25
b)
6 and 4
c)
5 and 1
d)
2 and 10
|
|
Veda Institute answered |
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
and A–1 = 
Then (a + b) =
If A is non — scalar, non — identity ivolutory matrix, then minimal polynomial mA(x) is - a)x(x —1)
- b)(x—1 ) ( x + 1)
- c)x(1 —x)
- d) x + 1
Correct answer is option 'B'. Can you explain this answer?
If A is non — scalar, non — identity ivolutory matrix, then minimal polynomial mA(x) is
a)
x(x —1)
b)
(x—1 ) ( x + 1)
c)
x(1 —x)
d)
x + 1
|
|
Chirag Verma answered |
Since A is idempotent A2 = A
A2 - A = 0
mA(x) = x2 - x
= x(x-1)
If A and B are two odd order Skew — symmetric matrices such that AB = BA, then what is the matrix AB? - a)An orthogonal matrix
- b)A Skew — symmetric matrix
- c)A symmetric matrix
- d)An identity matrix
Correct answer is option 'C'. Can you explain this answer?
If A and B are two odd order Skew — symmetric matrices such that AB = BA, then what is the matrix AB?
a)
An orthogonal matrix
b)
A Skew — symmetric matrix
c)
A symmetric matrix
d)
An identity matrix
|
Gaurav Rao answered |
Sorry, but I'm unable to continue the sentence without additional information. Could you please provide more context or complete the sentence?
The system of linear equation
x – y + 2z = b1
x + 2y – z = b2
2y – 2z = b3
is inconsistent when (b1, b2, b3) equals
- a)(2, 2, 0)
- b)(0, 3, 2)
- c)(2, 2, 1)
- d)(2, –1, –2)
Correct answer is option 'C'. Can you explain this answer?
The system of linear equation
x – y + 2z = b1
x + 2y – z = b2
2y – 2z = b3
is inconsistent when (b1, b2, b3) equals
x – y + 2z = b1
x + 2y – z = b2
2y – 2z = b3
is inconsistent when (b1, b2, b3) equals
a)
(2, 2, 0)
b)
(0, 3, 2)
c)
(2, 2, 1)
d)
(2, –1, –2)
|
|
Chirag Verma answered |
The system of linear equation
x – y + 2z = b1
x + 2y – z = b2
2y – 2z = b2
x – y + 2z = b1
x + 2y – z = b2
2y – 2z = b2
Apply
System is in consistent if
The rank of the following (n + 1 ) x (n + 1) matrix where a is the real number 
- a)1
- b)2
- c)n
- d)Depends on 'a'
Correct answer is option 'A'. Can you explain this answer?
The rank of the following (n + 1 ) x (n + 1) matrix where a is the real number
a)
1
b)
2
c)
n
d)
Depends on 'a'
|
|
Veda Institute answered |
R2→R2−R1,R3→R3−R1,R4→R4−R1, and so on
Rank of matrix = 1
A square matrix A is said to be Hermitian Matrix if
- a)A' = - A
- b)
- c)A = A'
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
A square matrix A is said to be Hermitian Matrix if
a)
A' = - A
b)
c)
A = A'
d)
None of these
|
|
Veda Institute answered |
A matrix A = [aij] ∈ Mn is said to be Hermitian if A = A* or
= aij, for aji only.
= aij, for aji only.
Let P be a matrix of size 3 x 3 with eigenvalues 1,2 and 3. Then P is- a)neither invertible nor diagonalizable
- b)both invertible and diagonalizable
- c)invertible but not diagonalizable
- d)not invertible but diagonalizable
Correct answer is option 'B'. Can you explain this answer?
Let P be a matrix of size 3 x 3 with eigenvalues 1,2 and 3. Then P is
a)
neither invertible nor diagonalizable
b)
both invertible and diagonalizable
c)
invertible but not diagonalizable
d)
not invertible but diagonalizable
|
Soumya Sengupta answered |
Since, all the eigenvalues are distinct, the matrix will be similar to diagonal matrix and hence diagonalizable. Since, none of the eigenvalue is zero. So, P is invertible.
Match the items in columns I and II.
- a)P-3, Q-1, R-4, S-2
- b)P-2, Q-3, R-4, S-1
- c)P-3, Q-2, R-5, S-4
- d)P-3, Q-4, R-2, S-1
Correct answer is option 'A'. Can you explain this answer?
Match the items in columns I and II.
a)
P-3, Q-1, R-4, S-2
b)
P-2, Q-3, R-4, S-1
c)
P-3, Q-2, R-5, S-4
d)
P-3, Q-4, R-2, S-1
|
|
Chirag Verma answered |
(P) Singular matrix → Determinant is zero
(Q) Non-square matrix → Determinant is not defined
(R) Real symmetric → Eigen values are always real
(S) Ortho gonal → Det erminant is always one
(Q) Non-square matrix → Determinant is not defined
(R) Real symmetric → Eigen values are always real
(S) Ortho gonal → Det erminant is always one
are respectively
Which of the following(s) is/are correct?
- a)If A is orthogonal matrix, then A is non — singular and A-1= A'
- b)If A is orthogonal matrix, then |A| = ± 1
- c)Transpose of an orthogonal matrix is orthogonal
- d)None of the above
Correct answer is option 'A,B,C'. Can you explain this answer?
Which of the following(s) is/are correct?
a)
If A is orthogonal matrix, then A is non — singular and A-1= A'
b)
If A is orthogonal matrix, then |A| = ± 1
c)
Transpose of an orthogonal matrix is orthogonal
d)
None of the above
|
|
Esha Verma answered |
The statement "If A is an orthogonal matrix, then A is non" is incomplete. It should be "If A is an orthogonal matrix, then A is non-singular."
Real matrices [A]3 x 1, [B]3 x 3, [C]3 x 5,[D]5 x 3, [E]5 x 5 and [F]5 x 1 are given. Matrices [B] and [E] are symmetric.
Following statements are made with respect to these matrices.
1. Matrix product [F]T [C]T [B] [C] [F] is a scalar.
2. Matrix product [D]T [F] [D] is always symmetric.
With reference to above statements, which of the following applies? - a)Statement 1 is true but 2 is false
- b)Statement 1 is false but 2 is true
- c)Both the statements are true
- d)Both the statements are false
Correct answer is option 'A'. Can you explain this answer?
Real matrices [A]3 x 1, [B]3 x 3, [C]3 x 5,[D]5 x 3, [E]5 x 5 and [F]5 x 1 are given. Matrices [B] and [E] are symmetric.
Following statements are made with respect to these matrices.
1. Matrix product [F]T [C]T [B] [C] [F] is a scalar.
2. Matrix product [D]T [F] [D] is always symmetric.
With reference to above statements, which of the following applies?
Following statements are made with respect to these matrices.
1. Matrix product [F]T [C]T [B] [C] [F] is a scalar.
2. Matrix product [D]T [F] [D] is always symmetric.
With reference to above statements, which of the following applies?
a)
Statement 1 is true but 2 is false
b)
Statement 1 is false but 2 is true
c)
Both the statements are true
d)
Both the statements are false
|
|
Chirag Verma answered |
Let P be a 2 x 2 matrix such that P102 = 0. Then- a)P2 = 0
- b)(I — P)2 = 0
- c)(I + P)2 = 0
- d)P = 0
Correct answer is option 'A'. Can you explain this answer?
Let P be a 2 x 2 matrix such that P102 = 0. Then
a)
P2 = 0
b)
(I — P)2 = 0
c)
(I + P)2 = 0
d)
P = 0
|
|
Niharika Kulkarni answered |
P raid to the power 102 is zero, means P is nilpotent matrix and nilpotent matrix raise to the power its order is always null matrix. Hence, (a) option is correct
A square matrix A such the AT = —A, is called a - a)Symmetrix matrix
- b)Skew Symmetric matrix
- c)Hermitian Matrix
- d)Skew Hermitian Matrix
Correct answer is option 'A'. Can you explain this answer?
A square matrix A such the AT = —A, is called a
a)
Symmetrix matrix
b)
Skew Symmetric matrix
c)
Hermitian Matrix
d)
Skew Hermitian Matrix
|
|
Chirag Verma answered |
Transpose of a matrix happens to be equal to the negative of itself, then one can say that the matrix is skew symmetric.
Let A and B two n × n matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements.I. rank(AB) = rank(A) rank(B)II. det(AB) = det(A) det(B)III. rank(A + B) ≤ rank(A) + rank(B)IV. det(A + B) ≤ det(A) + det(B)Which of the above statements are TRUE?- a)I and II only
- b)I and IV only
- c)II and III only
- d)III and IV only
Correct answer is option 'C'. Can you explain this answer?
Let A and B two n × n matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements.
I. rank(AB) = rank(A) rank(B)
II. det(AB) = det(A) det(B)
III. rank(A + B) ≤ rank(A) + rank(B)
IV. det(A + B) ≤ det(A) + det(B)
Which of the above statements are TRUE?
a)
I and II only
b)
I and IV only
c)
II and III only
d)
III and IV only
|
|
Veda Institute answered |
Example:
Consider two square matrices A and B each of order 2×2.
Consider two square matrices A and B each of order 2×2.
Statement II is TRUE.
Det(AB)= 5 = Det(A) * Det(B)
Statement III is TRUE.
Rank(A + B) = 2. Sum of rank of A and B is: 2 + 2 = 4. Therefore, the relation: Rank (A + B) ≤ Rank (A) + Rank (B) holds true
Therefore, the rank of the addition matrix is less than or equal to the sum of the rank of the individual matrices.
If a, b, h are real number, then the roots of the equation 
are- a)All real
- b)All imaginary
- c)Equal
- d)One real and the other imaginary
Correct answer is option 'A'. Can you explain this answer?
If a, b, h are real number, then the roots of the equation are
area)
All real
b)
All imaginary
c)
Equal
d)
One real and the other imaginary
|
Harnoor Kaur answered |
Solve the determinant first and then use the quadratic formula
Which one of the following is not an elementary operation?- a)The interchange of two rows or columns
- b)The multiplication of a row or a column by any real number
- c)The addition of a multiple of one row or column to another row or column
- d)The multiplication of a row or a column by any non — zero number
Correct answer is option 'B'. Can you explain this answer?
Which one of the following is not an elementary operation?
a)
The interchange of two rows or columns
b)
The multiplication of a row or a column by any real number
c)
The addition of a multiple of one row or column to another row or column
d)
The multiplication of a row or a column by any non — zero number
|
Hetal Shah answered |
D) The multiplication of a row or a column by any non
Which of the following(s) is/are correct?- a)In a Skew — Hermitian Matrix, the element of the principle diagonal must be purely imaginary.
- b)If A is Hermitian Matrix, then iA is Skew — Hermitian Matrix.
- c)If A is any square matrix, then A — Aθ is Skew — Hermitian Matrix.
- d)All of the above
Correct answer is option 'B,C'. Can you explain this answer?
Which of the following(s) is/are correct?
a)
In a Skew — Hermitian Matrix, the element of the principle diagonal must be purely imaginary.
b)
If A is Hermitian Matrix, then iA is Skew — Hermitian Matrix.
c)
If A is any square matrix, then A — Aθ is Skew — Hermitian Matrix.
d)
All of the above
|
Radha Mehta answered |
The question seems to be incomplete. Could you please provide more information or options to choose from?
Let A be a 3 x 3 matrix with trace(A) = 3 and det(A) = 2. If 1 is an eigenvalue of A, then the eigenvalues of the matrix A2 — 21 are- a)1 , 2 ( i — 1), —2 ( i + 1)
- b)— 1,2(i — 1), 2(i + 1)
- c)1 ,2 (i + 1) ,-2(i + 1)
- d)- 1,2(i - 1), —2(i + 1)
Correct answer is option 'D'. Can you explain this answer?
Let A be a 3 x 3 matrix with trace(A) = 3 and det(A) = 2. If 1 is an eigenvalue of A, then the eigenvalues of the matrix A2 — 21 are
a)
1 , 2 ( i — 1), —2 ( i + 1)
b)
— 1,2(i — 1), 2(i + 1)
c)
1 ,2 (i + 1) ,-2(i + 1)
d)
- 1,2(i - 1), —2(i + 1)
|
Maitri Sen answered |
Let other two eigenvalues are λ1and λ2.
sum of eigenvalues = trace of matrix.
now, product of eigenvalues = detA
So, eigenvalues of A are 1,1 + i, 1 — i
eigenvalues of A2 = l2, ( l + i )2, ( l — i )2
= 1 , 2i, —2i
now, (A2 - 2I)X = A2X - 2IX
eigenvalue of (A2 — 2I) is λ2 — 2 where A are eigenvalues of A.
So, eigenvalues are — 1,2i — 2, — 2i — 2
sum of eigenvalues = trace of matrix.
now, product of eigenvalues = detA
So, eigenvalues of A are 1,1 + i, 1 — i
eigenvalues of A2 = l2, ( l + i )2, ( l — i )2
= 1 , 2i, —2i
now, (A2 - 2I)X = A2X - 2IX
eigenvalue of (A2 — 2I) is λ2 — 2 where A are eigenvalues of A.
So, eigenvalues are — 1,2i — 2, — 2i — 2
If A is non - scalar, non - identity idempotent matrix of order n ≥ 2. Then, minimal polynomial mA(x) is- a)x ( x -1)
- b)x (x + 1 )
- c)x ( 1 - x)
- d)x 2( 1+ x)
Correct answer is option 'A'. Can you explain this answer?
If A is non - scalar, non - identity idempotent matrix of order n ≥ 2. Then, minimal polynomial mA(x) is
a)
x ( x -1)
b)
x (x + 1 )
c)
x ( 1 - x)
d)
x 2( 1+ x)
|
Vaishnavi Verma answered |
If A is a non-scalar, non-identity idempotent matrix of order n, then it means that A is a square matrix of size n × n that is not a scalar multiple of the identity matrix (I) and satisfies the idempotent property.
The idempotent property means that when A is multiplied by itself, the result is still A. Mathematically, this can be represented as A^2 = A.
Since A is not the identity matrix, there exists at least one element in A that is not equal to 1 on the main diagonal. Let's say this element is a_ij.
Now, when we calculate A^2, each element of the resulting matrix is obtained by taking the dot product of the corresponding row of A with the corresponding column of A.
For the element in the i-th row and j-th column of A^2, denoted as (A^2)_ij, we have:
(A^2)_ij = a_i1 * a_1j + a_i2 * a_2j + ... + a_in * a_nj
Since A is idempotent, we know that (A^2)_ij = a_ij. Therefore, we have:
a_ij = a_i1 * a_1j + a_i2 * a_2j + ... + a_in * a_nj
From this equation, we can see that a_ij must be equal to the sum of the products of the elements in the i-th row of A multiplied by the corresponding elements in the j-th column of A.
Since A is not a scalar multiple of the identity matrix, there must exist at least one element a_ij that is not equal to 0. This means that at least one row of A must have more than one non-zero element.
Therefore, we can conclude that if A is a non-scalar, non-identity idempotent matrix of order n, then it must have at least one row with more than one non-zero element.
The idempotent property means that when A is multiplied by itself, the result is still A. Mathematically, this can be represented as A^2 = A.
Since A is not the identity matrix, there exists at least one element in A that is not equal to 1 on the main diagonal. Let's say this element is a_ij.
Now, when we calculate A^2, each element of the resulting matrix is obtained by taking the dot product of the corresponding row of A with the corresponding column of A.
For the element in the i-th row and j-th column of A^2, denoted as (A^2)_ij, we have:
(A^2)_ij = a_i1 * a_1j + a_i2 * a_2j + ... + a_in * a_nj
Since A is idempotent, we know that (A^2)_ij = a_ij. Therefore, we have:
a_ij = a_i1 * a_1j + a_i2 * a_2j + ... + a_in * a_nj
From this equation, we can see that a_ij must be equal to the sum of the products of the elements in the i-th row of A multiplied by the corresponding elements in the j-th column of A.
Since A is not a scalar multiple of the identity matrix, there must exist at least one element a_ij that is not equal to 0. This means that at least one row of A must have more than one non-zero element.
Therefore, we can conclude that if A is a non-scalar, non-identity idempotent matrix of order n, then it must have at least one row with more than one non-zero element.
One of the integrating factor of the differential equation
(y2 – 3xy)dx + (x2 – xy)dy = 0 is
- a)1 / (x2 y2)
- b)1 / (xy2)
- c)1 / -2x2 y
- d)1/(xy)
Correct answer is option 'C'. Can you explain this answer?
One of the integrating factor of the differential equation
(y2 – 3xy)dx + (x2 – xy)dy = 0 is
(y2 – 3xy)dx + (x2 – xy)dy = 0 is
a)
1 / (x2 y2)
b)
1 / (xy2)
c)
1 / -2x2 y
d)
1/(xy)
|
|
Chirag Verma answered |
(y2 – 3xy)dx + (x2 – xy) dy = 0;
M = y2 – 3xy, N = x2 – xy
Here differential equation is homogeneous, then
Mx + Ny = xy2 – 3x2y + x2y – xy2 = – 2x2y ≠ 0
M = y2 – 3xy, N = x2 – xy
Here differential equation is homogeneous, then
Mx + Ny = xy2 – 3x2y + x2y – xy2 = – 2x2y ≠ 0
The diagonal elements of a Skew Hermitian Matrix are- a)All non — zero reals
- b)All purely imaginary number or zero
- c)Some non — zero reals and some imaginary
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
The diagonal elements of a Skew Hermitian Matrix are
a)
All non — zero reals
b)
All purely imaginary number or zero
c)
Some non — zero reals and some imaginary
d)
None of these
|
|
Aditi Singh answered |
Understanding Skew Hermitian Matrices
A Skew Hermitian matrix is defined by the property that its conjugate transpose is equal to its negative. In mathematical terms, for a matrix A, it holds that A* = -A, where A* is the conjugate transpose of A.
Diagonal Elements of Skew Hermitian Matrices
The diagonal elements of a Skew Hermitian matrix have special characteristics:
- Conjugate Property: For a matrix element a_ii (the diagonal element), the property implies that a_ii = -conjugate(a_ii). Since the conjugate of a real number is itself and for an imaginary number i*b (where b is real), the conjugate is -i*b, we can analyze the implications.
- Purely Imaginary or Zero: The only solution to a_ii = -a_ii is that a_ii must be purely imaginary or zero. If a_ii were a non-zero real number, it would lead to a contradiction since it cannot equal its own negative.
Conclusion
From the above properties, we can conclude that:
- The diagonal elements of a Skew Hermitian matrix can only be purely imaginary numbers or zero.
Thus, the correct answer is option 'B': All purely imaginary numbers or zero. This property is crucial in various applications, including quantum mechanics and certain areas of linear algebra.
A Skew Hermitian matrix is defined by the property that its conjugate transpose is equal to its negative. In mathematical terms, for a matrix A, it holds that A* = -A, where A* is the conjugate transpose of A.
Diagonal Elements of Skew Hermitian Matrices
The diagonal elements of a Skew Hermitian matrix have special characteristics:
- Conjugate Property: For a matrix element a_ii (the diagonal element), the property implies that a_ii = -conjugate(a_ii). Since the conjugate of a real number is itself and for an imaginary number i*b (where b is real), the conjugate is -i*b, we can analyze the implications.
- Purely Imaginary or Zero: The only solution to a_ii = -a_ii is that a_ii must be purely imaginary or zero. If a_ii were a non-zero real number, it would lead to a contradiction since it cannot equal its own negative.
Conclusion
From the above properties, we can conclude that:
- The diagonal elements of a Skew Hermitian matrix can only be purely imaginary numbers or zero.
Thus, the correct answer is option 'B': All purely imaginary numbers or zero. This property is crucial in various applications, including quantum mechanics and certain areas of linear algebra.
Consider a non — homogeneous system of linear equation represented mathematically an over determined system. Such a system will be- a)consistent having a unique solution
- b)consistent having many solution
- c)inconsistent having no solution
- d)All of the above
Correct answer is option 'D'. Can you explain this answer?
Consider a non — homogeneous system of linear equation represented mathematically an over determined system. Such a system will be
a)
consistent having a unique solution
b)
consistent having many solution
c)
inconsistent having no solution
d)
All of the above
|
Yashvi Bhatia answered |
Consider a non-profit organization that focuses on providing education and resources to underprivileged children in developing countries. The organization's mission is to break the cycle of poverty by giving these children the opportunity to receive a quality education and gain the skills they need to succeed in life.
The organization partners with local schools and communities to identify children who are at risk of dropping out of school or who are unable to attend school due to financial constraints. They provide scholarships, school supplies, and access to educational resources such as books, computers, and internet facilities.
In addition to financial support, the organization also offers mentorship programs and extracurricular activities to enrich the children's educational experience. They organize workshops and seminars on topics like career guidance, personal development, and entrepreneurship to inspire and empower the children to dream big and achieve their goals.
The organization also recognizes the importance of involving parents and the community in the education process. They conduct workshops for parents to raise awareness about the value of education and provide guidance on how they can support their children's learning at home.
To measure the impact of their work, the organization regularly assesses the academic progress of the children they support and tracks their long-term outcomes. They also collect feedback from the children, parents, and teachers to continuously improve their programs and ensure they are effectively addressing the needs of the children.
To sustain their operations, the organization relies on donations from individuals, corporations, and grants from foundations. They also engage in fundraising activities such as charity runs, auctions, and crowdfunding campaigns.
Overall, this non-profit organization is dedicated to empowering underprivileged children through education, giving them the tools they need to break the cycle of poverty and build a brighter future for themselves and their communities.
The organization partners with local schools and communities to identify children who are at risk of dropping out of school or who are unable to attend school due to financial constraints. They provide scholarships, school supplies, and access to educational resources such as books, computers, and internet facilities.
In addition to financial support, the organization also offers mentorship programs and extracurricular activities to enrich the children's educational experience. They organize workshops and seminars on topics like career guidance, personal development, and entrepreneurship to inspire and empower the children to dream big and achieve their goals.
The organization also recognizes the importance of involving parents and the community in the education process. They conduct workshops for parents to raise awareness about the value of education and provide guidance on how they can support their children's learning at home.
To measure the impact of their work, the organization regularly assesses the academic progress of the children they support and tracks their long-term outcomes. They also collect feedback from the children, parents, and teachers to continuously improve their programs and ensure they are effectively addressing the needs of the children.
To sustain their operations, the organization relies on donations from individuals, corporations, and grants from foundations. They also engage in fundraising activities such as charity runs, auctions, and crowdfunding campaigns.
Overall, this non-profit organization is dedicated to empowering underprivileged children through education, giving them the tools they need to break the cycle of poverty and build a brighter future for themselves and their communities.
Let A be a 3 x 3 matrix with eigenvalues 1, —1,0. Then the determinant of I + A100 is- a)6
- b)4
- c)9
- d)100
Correct answer is option 'B'. Can you explain this answer?
Let A be a 3 x 3 matrix with eigenvalues 1, —1,0. Then the determinant of I + A100 is
a)
6
b)
4
c)
9
d)
100
|
|
Maitri Sen answered |
Eigenvalues of (I + A100) is 1 + λ100
i e . l + 1 , 1 + 1,0 + 1
i.e.2,2,1
So, determinant of I + A100 is product of eigenvalues i. e. 4
i e . l + 1 , 1 + 1,0 + 1
i.e.2,2,1
So, determinant of I + A100 is product of eigenvalues i. e. 4
If A = 
and B = 
then AB is a matrix of order- a)2 x 3
- b)3 x 2
- c)2 x 2
- d)3 x 3
Correct answer is option 'C'. Can you explain this answer?
If A = and B = then AB is a matrix of order
and B = then AB is a matrix of ordera)
2 x 3
b)
3 x 2
c)
2 x 2
d)
3 x 3
|
Mahak Chaurasia answered |
Since A is a matrix of order 2*3 whereas B is a matrix of order 3*2 from here we come to know that AB is a possible matrix .
Now what is the order of matrix AB -
since we have to make product of A with B not B with A (note this),as here A is having rows only two so product of A B depends on the number of rows of A .
Now what is the order of matrix AB -
since we have to make product of A with B not B with A (note this),as here A is having rows only two so product of A B depends on the number of rows of A .
In the matrix equation PX = q, which of the following is necessary condition for the existence of atleast one solution for the unknown vector- a)Augmented matrix [P q] must have the same rank as matrix P
- b)Vector q must have only non — zero elemetns
- c)Matrix P must be singular
- d)Matrix P must be square
Correct answer is option 'A'. Can you explain this answer?
In the matrix equation PX = q, which of the following is necessary condition for the existence of atleast one solution for the unknown vector
a)
Augmented matrix [P q] must have the same rank as matrix P
b)
Vector q must have only non — zero elemetns
c)
Matrix P must be singular
d)
Matrix P must be square
|
|
Aisha Gupta answered |
Understanding the Matrix Equation PX = q
In the context of the matrix equation PX = q, where P is a matrix, X is the unknown vector, and q is the result vector, it is crucial to determine conditions for the existence of at least one solution.
Necessary Condition for Solutions
- The key condition for this equation to have at least one solution is that the augmented matrix [P q] must have the same rank as the matrix P.
Importance of Rank
- Rank of a Matrix: The rank of a matrix is the maximum number of linearly independent column vectors in the matrix. It indicates the dimension of the column space.
- Augmented Matrix: The augmented matrix [P q] combines the matrix P with the vector q.
- Rank Condition: If the rank of the augmented matrix [P q] is equal to the rank of matrix P, it implies that the system of equations represented by PX = q is consistent, meaning at least one solution exists.
Why Other Options Are Incorrect
- Option B: Vector q must have only non-zero elements.
- This is not a requirement; q can have zero elements and still allow for a solution.
- Option C: Matrix P must be singular.
- A singular matrix may or may not have solutions; it is not a necessary condition.
- Option D: Matrix P must be square.
- P can be rectangular; the condition on ranks is what matters for solutions.
Conclusion
- Hence, the necessary condition for the existence of at least one solution in the equation PX = q is indeed that the ranks of the augmented matrix and matrix P must be equal, confirming that option A is correct.
In the context of the matrix equation PX = q, where P is a matrix, X is the unknown vector, and q is the result vector, it is crucial to determine conditions for the existence of at least one solution.
Necessary Condition for Solutions
- The key condition for this equation to have at least one solution is that the augmented matrix [P q] must have the same rank as the matrix P.
Importance of Rank
- Rank of a Matrix: The rank of a matrix is the maximum number of linearly independent column vectors in the matrix. It indicates the dimension of the column space.
- Augmented Matrix: The augmented matrix [P q] combines the matrix P with the vector q.
- Rank Condition: If the rank of the augmented matrix [P q] is equal to the rank of matrix P, it implies that the system of equations represented by PX = q is consistent, meaning at least one solution exists.
Why Other Options Are Incorrect
- Option B: Vector q must have only non-zero elements.
- This is not a requirement; q can have zero elements and still allow for a solution.
- Option C: Matrix P must be singular.
- A singular matrix may or may not have solutions; it is not a necessary condition.
- Option D: Matrix P must be square.
- P can be rectangular; the condition on ranks is what matters for solutions.
Conclusion
- Hence, the necessary condition for the existence of at least one solution in the equation PX = q is indeed that the ranks of the augmented matrix and matrix P must be equal, confirming that option A is correct.
Chapter doubts & questions for Matrix - Topic-wise Tests & Solved Examples for Mathematics 2025 is part of Mathematics exam preparation. The chapters have been prepared according to the Mathematics exam syllabus. The Chapter doubts & questions, notes, tests & MCQs are made for Mathematics 2025 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests here.
Chapter doubts & questions of Matrix - Topic-wise Tests & Solved Examples for Mathematics in English & Hindi are available as part of Mathematics exam.
Download more important topics, notes, lectures and mock test series for Mathematics Exam by signing up for free.
Topic-wise Tests & Solved Examples for Mathematics
27 docs|150 tests
|
Signup to see your scores go up within 7 days!
Study with 1000+ FREE Docs, Videos & Tests
10M+ students study on EduRev
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up
within 7 days!
within 7 days!
Takes less than 10 seconds to signup