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# If A and B are square matrices of the same order,then $(ii)\quad (kA)'=\text{________}.$(k is any scalar)

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Toolbox:
• If A_{i,j} be a matrix m*n matrix , then the matrix obtained by interchanging the rows and column of A is called as transpose of A.
Given:
If A and B are square matrices of the same order then
If K is any number real or complex and A be any matrix then
$(kA)'=kA'$
Proof:
Let A=$[a_{ij}]$
A' = $[a_{ji}]$
kA= $[ka_{ij}]$
(kA)'= $[ka_{ji}]$ $\rightarrow$ (1)
Now consider A' = $[a_{ji}]$
Multiply by scalar k on both side we get
kA'= $[ka_{ji}]$ $\rightarrow$ (2)
Hence from equation 1 and 2 we have
(kA)'=kA'