250+ TOP MCQs on Transpose of Matrices and Answers

Discrete Mathematics Problems on “Transpose of Matrices”.

1. For a matrix A, if a matrix B is obtained by changing its rows into columns and column into rows, then relation between A and B is?
a) A² = B
b) A^T = B
c) Depends on the matrix
d) None of the mentioned

Answer: b
Clarification: A = [a_ij] and B = [a_ji], B = A^T.

2. For matrix A, (A^T)^T is equals to ___________
a) A
b) A^T
c) Can’t say
d) None of the mentioned

Answer: a
Clarification: Transpose of a transposed matrix results in same matrix.

3. For matrix Aand a scalar k, (kA)^T is equal to _________
a) k(A)
b) k(A)^T
c) k²(A)
d) k²(A)^T

Answer: b
Clarification: Scalar has no effect on transpose.

4. If A is a lower triangular matrix then A^T is a _________
a) Lower triangular matrix
b) Upper triangular matrix
c) Null matrix
d) None of the mentioned

Answer: b
Clarification: By transpose a lower triangular matrix will turn to upper triangular matrix and vice – versa.

5. If matrix A and B are symmetric and AB = BA iff _________
a) AB is symmetric matrix
b) AB is an anti-symmetric matrix
c) AB is a null matrix
d) None of the mentioned

Answer: a
Clarification: For two symmetric matrices A and B, AB is a symmetric matrix if and only if AB = BA.

6. A matrix can be expressed as sum of symmetric and anti-symmetric matrices.
a) True
b) False

Answer: a
Clarification: Since A = (¹⁄₂)(A + A^T) + ((¹⁄₂)(A – A^T)

7. The determinant of a diagonal matrix is the product of leading diagonal’s element.
a) True
b) False

Answer: a
Clarification: Since in diagonal matrix all element other than diagonal are zero.

8. If for a square matrix A and B,null matrix O, (AB)^T = O implies A^T = O and B^T = O.
a) True
b) False

Answer: b
Clarification: Let A=[0 1 0 0 ], B=[1 0 0 0 ] AB=O and B, A^T, B^T is not equal to O.

9. Let A = [a_ij] given by ab_ij = (i-j)³ is a _________
a) Symmetric matrix
b) Anti-Symmetric matrix
c) Identity matrix
d) None of the mentioned

Answer: b
Clarification: aji =(j-i³) = -a_ij, A is Anti-symmetric matrix.

10. Trace of the matrix of odd ordered anti-symmetric matrix is _________
a) 0
b) 1
c) 2
d) All of the mentioned

Answer: a
Clarification: Since in odd ordered anti-symmetric matrix all diagonal matrix are zero.