Question

If A is a square matrix such that $A\left(\mathrm{adj}A\right)=\left[\begin{array}{ccc}5& 0& 0\\ 0& 5& 0\\ 0& 0& 5\end{array}\right]$, then write the value of |adj A|.

Answer 1

$$\begin{gathered} \mathrm{Given}:\hspace{0.17em}A\left(\mathrm{adj}A\right)=\left[500 \\ 050 \\ 005\right] \\ \Rightarrow \hspace{0.17em}\left|A\right|I_n=5\left[100 \\ 010 \\ 001\right] \\ \Rightarrow \hspace{0.17em}\left|A\right|=5 \\ \mathrm{Now},\left|\mathrm{adj}A\right|=\hspace{0.17em}{\left|A\right|}^{n-1}=5^{3-1}=25 \end{gathered}$$