Question

If $A=\left[\begin{array}{cc}\alpha & 0\\ 1& 1\end{array}\right]$ and $B=\left[\begin{array}{cc}1& 0\\ 5& 1\end{array}\right]$, then the value of $\alpha$ for which $A^2=B$ is:

• A. $1$
• B. $2$
• C. $4$
• D. No real values.

Answer 1

The correct option is D No real values.

Given, $A=\left[\begin{array}{cc}\alpha & 0\\ 1& 1\end{array}\right],B=\left[\begin{array}{cc}1& 0\\ 5& 1\end{array}\right]$ and $A^2=B$

$\Rightarrow A^2=\left[\begin{array}{cc}\alpha & 0\\ 1& 1\end{array}\right]\left[\begin{array}{cc}\alpha & 0\\ 1& 1\end{array}\right]=\left[\begin{array}{cc}{\alpha }^2& 0\\ \alpha +1& 1\end{array}\right]$

Now, $A^2=B$
$\Rightarrow \left[\begin{array}{cc}{\alpha }^2& 0\\ \alpha +1& 1\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 5& 1\end{array}\right]$
$\Rightarrow {\alpha }^2=1\text{and}\alpha +1=5$ which is not possible at the same time.
$\therefore$ No real values of $\alpha .$