Question

If A is a square matrix of order 2 such that A (adj A) = $\left[\begin{array}{cc}10& 0\\ 0& 10\end{array}\right],\mathrm{then}\left|A\right|$ =______________.

Answer 1

As we know that, A(adj A) = |A|I.

But it is given that A (adj A) = $\left[\begin{array}{cc}10& 0\\ 0& 10\end{array}\right]$
$$\begin{gathered} \Rightarrow A\left(\mathrm{adj}A\right)=10\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right] \\ \Rightarrow A\left(\mathrm{adj}A\right)=10I \\ \Rightarrow \left|A\right|I=10I \\ \Rightarrow \left|A\right|=10 \end{gathered}$$

Hence, |A| = 10.