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If adj $A=\left[\begin{array}{cc}a & 0 \\ -1 & b\end{array}\right]$ and $a b \neq 0$, then what is the value of $\left|A^{-1}\right| ?$
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For $2 \times 2$ matrix,
$|A|=|\operatorname{adj} A|$
$\quad=(a b-0)=a b$
$\therefore A^{-1}=\frac{a d j A}{|A|}=\frac{1}{a b} \cdot\left(\begin{array}{cc}a & 0 \\ -1 & b\end{array}\right)$
$\left|A^{-1}\right|=\frac{1}{a b}(a b)=1$
$|A|=|\operatorname{adj} A|$
$\quad=(a b-0)=a b$
$\therefore A^{-1}=\frac{a d j A}{|A|}=\frac{1}{a b} \cdot\left(\begin{array}{cc}a & 0 \\ -1 & b\end{array}\right)$
$\left|A^{-1}\right|=\frac{1}{a b}(a b)=1$
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