Question

A and B are square matrices of the same order, then __________.
(i) (AB)^T = __________
(ii) (KA)^T = __________
(iii) (k (A – B))^T = __________
where k is any scalar.

Answer 1

It is given that, A and B are square matrices of the same order.

(i) (AB)^T = $\underline{B^T{A}^T}$

(ii) (kA)^T = $\underline{k{\left(A\right)}^T}$, where k is any scalar

(iii) [k (A – B)]^T

$=k{\left(A-B\right)}^T$ [(kA)^T = k(A)^T]

$=k\left(A^T-B^T\right)$ [(A + B)^T = A^T + B^T]

∴ [k (A – B)]^T = $\underline{k\left(A^T-B^T\right)}$, where k is any scalar