Question

If A is a 3 × 3 invertible matrix, then what will be the value of k if det(A^–1) = (det A)^k.

Answer 1

As we know that

$$\begin{gathered} A^{-1}=\frac{\mathrm{Adj}A}{\left|\mathrm{A}\right|} \\ \therefore \left|A^{-1}\right|=\frac{\left|\mathrm{Adj}A\right|}{\left|A\right|} \\ =\frac{{\left|A\right|}^{3-1}}{\left|A\right|}\left[\because \mathrm{If}A\mathrm{is}\mathrm{a}\mathrm{non}\mathrm{singular}\mathrm{matrix}\mathrm{of}\mathrm{order}n,\mathrm{then}\left|\mathrm{adj}\left(A\right)\right|={\left|\mathrm{A}\right|}^{n-1}\right] \end{gathered}$$
​$$\begin{gathered} =\frac{{\left|A\right|}^2}{\left|A\right|} \\ =\left|A\right| \end{gathered}$$
$\mathrm{As}\mathrm{we}\mathrm{are}\mathrm{given}\mathrm{that}\left|A^{-1}\right|={\left|A\right|}^k$
$\therefore k=1$