Question

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

• A. $\pm 3$
• B. $\pm 2$
• C. $\pm 5$
• D. $0$

Answer 1

The correct option is A $\pm 3$

We have,

$A=\left[\begin{array}{cc}\alpha & 2\\ 2& \alpha \end{array}\right]\left|A^3\right|=125{\left|A\right|}^3=125A=5$

$Now,\left|A\right|=\left|\begin{array}{cc}\alpha & 2\\ 1& \alpha \end{array}\right|5={\alpha }^2-4{\alpha }^2-4=5\alpha =\pm 3$

$Hence,theoptionAisthecorrectanswer.$