If A is a 3 × 3 invertible matrix,


If $A$ is a $3 \times 3$ invertible matrix, then what will be the value of $k$ if $\operatorname{det}\left(A^{-1}\right)=(\operatorname{det} A)^{k}$.


As we know that

$A^{-1}=\frac{\operatorname{Adj} A}{|\mathrm{~A}|}$

$\therefore\left|A^{-1}\right|=\frac{|\operatorname{Adj} A|}{|A|}$

$=\frac{|A|^{2-1}}{|A|} \quad\left[\because\right.$ If $A$ is a non singular matrix of order $n$, then $\left.|\operatorname{adj}(A)|=|\mathrm{A}|^{n-1}\right]$


As we are given that $\left|A^{-1}\right|=\mid A^{k}$

$\therefore k=1$

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