Question

If A is a square matrix with $\left|A\right|$ = 4 then find the value of $\left|A.(adjA)\right|.$

Answer 1

Given:
A is a square matrix
$\left|A\right|$ = 4


Now,
$$\begin{gathered} A\left(\mathrm{adj}A\right)=\left|A\right|I \\ \Rightarrow \left|A\left(\mathrm{adj}A\right)\right|=\left|\left|A\right|I\right| \\ \Rightarrow \left|A\left(\mathrm{adj}A\right)\right|={\left|A\right|}^n\left|I\right|,\mathrm{where}n\mathrm{is}\mathrm{the}\mathrm{order}\mathrm{of}I \\ \Rightarrow \left|A\left(\mathrm{adj}A\right)\right|={\left|A\right|}^n\times 1 \\ \Rightarrow \left|A\left(\mathrm{adj}A\right)\right|=4^n\left(\because \left|A\right|=4\right) \end{gathered}$$


Hence, $\left|A\left(\mathrm{adj}A\right)\right|=\underline{4^n}.$