Question

If A is an invertible matrix of order 3 and $\left|A\right|=3,\mathrm{then}\left|adjA\right|$ = ___________________.

Answer 1

Given:
A is an invertible matrix of order 3
$\left|A\right|$ = 3


As we know,
$$\begin{gathered} \left|\mathrm{adj}A\right|={\left|A\right|}^{n-1},\mathrm{where}n\mathrm{is}\mathrm{the}\mathrm{order}\mathrm{of}A \\ \Rightarrow \left|\mathrm{adj}A\right|={\left|A\right|}^{3-1}\left(\because \mathrm{Order}\mathrm{of}A\mathrm{is}3\right) \\ \Rightarrow \left|\mathrm{adj}A\right|={\left|A\right|}^2 \\ \Rightarrow \left|\mathrm{adj}A\right|={\left(3\right)}^2\left(\because \left|A\right|=3\right) \\ \Rightarrow \left|\mathrm{adj}A\right|=9 \end{gathered}$$


Hence, |adj A| = 9.