Question:
$A$ and $B$ are square matrices of the same order, then______
(i) $(A B)^{\top}=$________
(ii) $(K A)^{\top}=$_______
(iii) $(k(A-\bar{B}))^{\top}=$_______
where $k$ is any scalar.
Solution:
It is given that, $A$ and $B$ are square matrices of the same order.
(i) $(A B)^{\top}=\quad B^{T} A^{T}$
(ii) $(k A)^{\top}=k(A)^{T}$, where $k$ is any scalar
(iii) $[K(A-B)]^{T}$
$=k(A-B)^{T} \quad\left[(k A)^{T}=k(A)^{T}\right]$
$=k\left(A^{T}-B^{T}\right) \quad\left[(A+B)^{T}=A^{T}+B^{\top}\right]$
$\therefore[k(A-B)]^{T}=k\left(A^{T}-B^{T}\right)$, where $k$ is any scalar
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