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If $A=\left[\begin{array}{ccc}1 & 2 & 3 \\ -1 & 1 & 2 \\ 1 & 2 & 4\end{array}\right]$, and $A(\operatorname{adj} A)=k I$, then the value of $(k+1)^4$ is
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The correct answer is:
256
$|A|=\left|\begin{array}{ccc}
1 & 2 & 3 \\
-1 & 1 & 2 \\
1 & 2 & 4
\end{array}\right|=1(0)-2(-6)+3(-3)=3$
We know that $\mathrm{A}(\operatorname{adj} \mathrm{A})=|\mathrm{A}| \mathrm{I}$
$\begin{aligned}
& \therefore \mathrm{A}(\operatorname{adj} \mathrm{A})=3 \mathrm{I} \Rightarrow \mathrm{k}=3 \\
& (\mathrm{k}+1)^4=(3+1)^4=256
\end{aligned}$
1 & 2 & 3 \\
-1 & 1 & 2 \\
1 & 2 & 4
\end{array}\right|=1(0)-2(-6)+3(-3)=3$
We know that $\mathrm{A}(\operatorname{adj} \mathrm{A})=|\mathrm{A}| \mathrm{I}$
$\begin{aligned}
& \therefore \mathrm{A}(\operatorname{adj} \mathrm{A})=3 \mathrm{I} \Rightarrow \mathrm{k}=3 \\
& (\mathrm{k}+1)^4=(3+1)^4=256
\end{aligned}$
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