Question

If $A=\left[\begin{array}{cc}a& b\\ b& a\end{array}\right]$ and $A^2=\left[\begin{array}{cc}\alpha & \beta \\ \beta & \alpha \end{array}\right]$, then

• A. $\alpha =a^2+b^2,\beta =2ab$
• B. $\alpha =a^2+b^2,\beta =a^2-b^2$
• C. $\alpha =2ab,\beta =a^2+b^2$
• D. $\alpha =a^2+b^2,\beta =ab$

Answer 1

The correct option is A $\alpha =a^2+b^2,\beta =2ab$

We are given $A=\left[\begin{array}{cc}a& b\\ b& a\end{array}\right]$

Now we find $A^2=AA=\left[\begin{array}{cc}a& b\\ b& a\end{array}\right]\left[\begin{array}{cc}a& b\\ b& a\end{array}\right]$

${\Rightarrow A}^2=\left[\begin{array}{cc}{a}^2+b^2& 2ab\\ 2ab& a^2+b^2\end{array}\right]$

Compare it with the given $A^2$ matrix. we get

$\alpha =a^2+b^2,\beta =2ab$