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

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

Answer 1

Answer: (C)

$\frac{1}{5}$

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

$\therefore A^2=\left[\begin{array}{ccc}5& 5\alpha & \alpha \\ 0& \alpha & 5\alpha \\ 0& 0& 5\end{array}\right]\left[\begin{array}{ccc}5& 5\alpha & \alpha \\ 0& \alpha & 5\alpha \\ 0& 0& 5\end{array}\right]=\left[\begin{array}{ccc}25& 25\alpha +5{\alpha }^2& 10\alpha +25{\alpha }^2\\ 0& {\alpha }^2& 5a^2+25\\ 0& 0& 25\end{array}\right]$

$\left|A^2\right|=25\left|\begin{array}{cc}25& 25\alpha +{\alpha }^2\\ 0& {\alpha }^2\end{array}\right|=625{\alpha }^2$

But $\left|A^2\right|=25$

$\therefore 625{\alpha }^2=25\Rightarrow \left|\alpha \right|=\frac{1}{5}$