Question

If $A=\begin{bmatrix}a & b \\ b & a\end{bmatrix}$ and $A^{2}=\begin{bmatrix}\alpha & \beta \\ \beta & \alpha\end{bmatrix}$, then

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

Answer 1

Answer: (A)

$A^{2}=\begin{bmatrix}a & b \\ b & a\end{bmatrix}\begin{bmatrix}a & b \\ b & a\end{bmatrix}$
$=\begin{bmatrix}a^{2}+b^{2} & 2 a b \\ 2 a b & a^{2}+b^{2}\end{bmatrix}=\begin{bmatrix}\alpha & \beta \\ \beta & \alpha\end{bmatrix}$ (given)
$\Rightarrow \alpha=a^{2}+b^{2}, \beta=2 a b$