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If for any $2 \times 2$ square matrix $A$,$A(\operatorname{adj} A)=\left[\begin{array}{ll}8 & 0 \\ 0 & 8\end{array}\right]$, then find the value of $\operatorname{det}(A)$.
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8
Given, $A(\operatorname{adj} A)=\left[\begin{array}{ll}8 & 0 \\ 0 & 8\end{array}\right]$,
By using property $A(\operatorname{adj} A)=|A| \ln$ $\begin{array}{ll}\Rightarrow & |A| \ln =\left[\begin{array}{ll}8 & 0 \\ 0 & 8\end{array}\right] \\ \Rightarrow & |A| \ln =8\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] \Rightarrow|A|=8\end{array}$
By using property $A(\operatorname{adj} A)=|A| \ln$ $\begin{array}{ll}\Rightarrow & |A| \ln =\left[\begin{array}{ll}8 & 0 \\ 0 & 8\end{array}\right] \\ \Rightarrow & |A| \ln =8\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] \Rightarrow|A|=8\end{array}$
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