Question

Let $A=\left[\begin{array}{ccc}-1& -2& -2\\ 2& 1& -2\\ 2& -2& 1\end{array}\right].$ If adj. $A=k{A}^T$ then the value of ' $k$ ' is

Answer 1

Answer: 3

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