Question

For any 2 × 2 matrix, if $A\left(adjA\right)=\left[\begin{array}{cc}10& 0\\ 0& 10\end{array}\right]$, then |A| is equal to
(a) 20
(c) 100
(d) 10
(d) 0

Answer 1

$$\begin{gathered} \left(\mathrm{c}\right)10 \\ A\left(adjA\right)=\left[100 \\ 010\right] \\ \mathrm{By}\mathrm{definition},\mathrm{we}\mathrm{have} \\ A\left(adjA\right)=\left|A\right|I=\left(adjA\right)A(\mathrm{Where}I\mathrm{is}\mathrm{the}\mathrm{identity}\mathrm{matrix}) \\ \Rightarrow \left|A\right|I=A\left(adjA\right) \\ \Rightarrow \left|A\right|I=10\left[10 \\ 01\right] \\ \Rightarrow \left|A\right|=10 \end{gathered}$$