Question
If $A=\left[\begin{array}{ll}p & 2 \\ 2 & p\end{array}\right]$ and $\left|A^3\right|=125$, then find the value(s) of $p$.
✓
Answer
$\begin{array}{l}\text{Given,}\quad A=\left[\begin{array}{ll}p & 2 \\2 & p\end{array}\right] \text { and }\left|A^3\right|=125 \\\text { Since, } \quad\left|A^n\right|=|A|^n \\\therefore \quad\left|A^3\right|=|A|^3\end{array}$
$\text{Now,}\quad|A|=\left|\begin{array}{ll}p & 2 \\2 & p\end{array}\right|=p^2-4$
According to given condition,
$|A^{3}|=125$
$\Rightarrow |A|^{3}=125$
$\Rightarrow (p^{2}-4)^{3}=125$
$\Rightarrow (p^{2}-4)^{3}=5^{3}$
$\Rightarrow p^{2}-4=5$
$\Rightarrow p^{2}=9$
$\Rightarrow p=\pm3$
Hence, values of $p=\pm3.$
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