Question

If A is an invertible matrix of order 3 and $\left|A\right|=5,\mathrm{then}\mathrm{adj}\left(adjA\right)$ = __________________.

Answer 1

Given:
A is an invertible matrix of order 3
$\left|A\right|$ = 5


As we know,
$$\begin{gathered} \mathrm{adj}\left(\mathrm{adj}A\right)={\left|A\right|}^{n-2}A,\mathrm{where}n\mathrm{is}\mathrm{the}\mathrm{order}\mathrm{of}A \\ \Rightarrow \mathrm{adj}\left(\mathrm{adj}A\right)={\left|A\right|}^{3-2}A\left(\because \mathrm{Order}\mathrm{of}A\mathrm{is}3\right) \\ \Rightarrow \mathrm{adj}\left(\mathrm{adj}A\right)={\left|A\right|}^{1}A \\ \Rightarrow \mathrm{adj}\left(\mathrm{adj}A\right)=5A\left(\because \left|A\right|=5\right) \\ \Rightarrow \mathrm{adj}\left(\mathrm{adj}A\right)=5A \end{gathered}$$


Hence, adj (adj A) = 5A.