This set of Discrete Mathematics Problems focuses 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) A^{2} = B

b) A^{T} = B

c) Depends on the matrix

d) None of the mentioned

View Answer

Explanation: 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

View Answer

Explanation: 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) k^{2}(A)

d) k^{2}(A)^{T}

View Answer

Explanation: 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

View Answer

Explanation: 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

4) None of the mentioned

View Answer

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

6. State True or False:

A matrix can be expressed as sum of symmetric and anti -symmetric matrices.

a) True

b) False

View Answer

Explanation: Since A = (

^{1}⁄

_{2})(A + A

^{T}) + ((

^{1}⁄

_{2})(A – A

^{T})

7. State True or False:

The determinant of a diagonal matrix is the product of leading diagonal’s element.

a) True

b) False

View Answer

Explanation: Since in diagonal matrix all element other than diagonal are zero.

8. State whether the given statement is True or False.

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

View Answer

Explanation: 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

View Answer

Explanation: aji =(j-i

^{3}) = -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

View Answer

Explantion: Since in odd ordered anti-symmetric matrix all diagonal matrix are zero.

**Sanfoundry Global Education & Learning Series – Discrete Mathematics.**

To practice all areas of Discrete Mathematics Problems, __here is complete set of 1000+ Multiple Choice Questions and Answers__.