Q91 / 180 · KCET 2013 · Mathematics

If $A=\left[\begin{array}{cc}\alpha & 2 \\ 2 & \alpha\end{array}\right]$ and $\left|A^{3}\right|=125$, then $\alpha$ is equal to

  • A
    $\pm 1$
  • B
    $\pm 2$
  • Correct answer
    C
    $\pm 3$
  • D
    $\pm 5$

Solution

Given, $\quad A=\left[\begin{array}{ll}\alpha & 2 \\ 2 & \alpha\end{array}\right]$ Now, $\quad|A|=\left|\begin{array}{cc}\alpha & 2 \\ 2 & \alpha\end{array}\right|=\alpha^{2}-4...(i)$ Also, given, $\begin{array}{ll} & \left|A^{3}\right|=125 \\ \Rightarrow \quad & |A|^{3}=125=(5)^{3} \quad\left(\because\left|A^{n}\right|=|A|^{n}\right) \\ \therefore & |A|=5 \end{array}$ On putting this value in Eq. (i), we get $\begin{aligned} 5 &=\alpha^{2}-4 \\ \Rightarrow \quad \alpha^{2} &=9=(3)^{2} \\ \Rightarrow \quad \alpha &=\pm 3 \end{aligned}$

Hence, the correct answer is option: 'C'.

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