Question:
For any $2 \times 2$ matrix, if $A(\operatorname{adj} A)=\left[\begin{array}{cc}10 & 0 \\ 0 & 10\end{array}\right]$, then $|A|$ is equal to
(a) 20
(c) 100
(d) 10
(d) 0
Solution:
(c) 10
$A(a d j A)=\left[\begin{array}{ll}10 & 0\end{array}\right.$
$\left.\begin{array}{ll}0 & 10\end{array}\right]$
By definition, we have
$A(\operatorname{adj} A)=|A| I=(\operatorname{adj} A) A$ (Where $I$ is the identity matrix)
$\Rightarrow|A| I=A(\operatorname{adj} A)$
$\Rightarrow|A| I=10\left[\begin{array}{ll}1 & 0\end{array}\right.$
$\left.\begin{array}{ll}0 & 1\end{array}\right]$
$\Rightarrow|A|=10$
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