Let $A = \begin{bmatrix} x + y & y \\ 2x & x - y \end{bmatrix}, B = \begin{bmatrix} 2 \\ -1 \end{bmatrix}$ and $C = \begin{bmatrix} 3 \\ 2 \end{bmatrix}$. If $AB = C$, then find $A^2$.
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → $\begin{bmatrix} 6 & -10 \\ 4 & 26 \end{bmatrix}$ ##
Here, $\begin{bmatrix} x+y & y \\ 2x & x-y \end{bmatrix} \begin{bmatrix} 2 \\ -1 \end{bmatrix} = \begin{bmatrix} 3 \\ 2 \end{bmatrix}$
$\Rightarrow \begin{bmatrix} 2x+y \\ 3x+y \end{bmatrix} = \begin{bmatrix} 3 \\ 2 \end{bmatrix}$
$2x + y = 3$
$3x + y = 2$
On solving above equations, we get $x = -1$ and $y = 5$.
$∴A = \begin{bmatrix} -1+5 & 5 \\ 2(-1) & -1-5 \end{bmatrix}$
$= \begin{bmatrix} 4 & 5 \\ -2 & -6 \end{bmatrix}$
Thus, $A^2 = \begin{bmatrix} 4 & 5 \\ -2 & -6 \end{bmatrix} \begin{bmatrix} 4 & 5 \\ -2 & -6 \end{bmatrix}$
$= \begin{bmatrix} 6 & -10 \\ 4 & -26 \end{bmatrix}$