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Determinants — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSDeterminants1 Mark
Question
For any matrix $A , A =\left[\begin{array}{cc}\alpha & -2 \\ -2 & \alpha\end{array}\right],\left| A ^3\right|=125$ then value of $\alpha$ is :
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Answer

(A) $\pm 3$
Here $A =\left[\begin{array}{cc}\alpha & -2 \\ -2 & \alpha\end{array}\right]$ and $\left| A ^3\right|=125$$
\begin{aligned}
\left|A^n\right|=|A|^n & \Rightarrow\left|A^3\right|=|A|^3 \\
& \Rightarrow 125=|A|^3 \\
& \Rightarrow 125=\left(\alpha^2-4\right)^3 \\
& \Rightarrow \alpha^2-4=5 \\
& \Rightarrow \alpha^2=9 \\
& \Rightarrow \alpha= \pm 3
\end{aligned}
$

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