Question

Let $A=\left[\begin{array}{ccc}5& 5\alpha & \alpha \\ 0& \alpha & 5\alpha \\ 0& 0& 5\end{array}\right]$ If $\left|A^2\right|=25,$ then $\left|\alpha \right|$ equals to

• A. $\frac{1}{5}$
• B. $5$
• C. $5^2$
• D. $1$

Answer 1

The correct option is A $\frac{1}{5}$

$A=\left[\begin{array}{ccc}5& 5\alpha & \alpha \\ 0& \alpha & 5\alpha \\ 0& 0& 5\end{array}\right]$

$|A|=25\alpha$
Given, $|A^2|=25$

$\Rightarrow |A{|}^2=25$

$\Rightarrow (25\alpha {)}^2=25$

$\Rightarrow {\alpha }^2=\frac{1}{25}$

$\Rightarrow |\alpha |=\frac{1}{5}$