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  4. Let A=[-1 -2 -2 2 1 -2 2 -2 1] . If adj. A=k AT then the value of ' k ' is

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Q. Let $A=\begin{bmatrix}-1 & -2 & -2 \\ 2 & 1 & -2 \\ 2 & -2 & 1\end{bmatrix} .$ If adj. $A=k A^{T}$ then the value of ' $k$ ' is

Matrices

Solution:

We know that $|{adj} A|=|A|^{2}$ for a $3 \times 3$ matrix
Given $({adj} A)=K A^{T} $
$\Rightarrow|{adj} A|=\left|K A^{T}\right|$
$=K^{3}|A|\left(\left|A^{T}\right|=|A|\right)$
$\therefore K^{3}|A|=|A|^{2}$
$ \Rightarrow K^{3} =|A| $;
Now det $A=-1(1-4)-2(-2-4)+2(4+2) $
$=27 \Rightarrow k^{3}=27 $
$\Rightarrow K=3$