Let matrix $A = \begin{vmatrix} 1 & -1 \\ 2 & 2 \end{vmatrix}$, $B = \begin{vmatrix} 1 1 & \alpha \\ \beta \beta & 1 \end{vmatrix}$ If $(A + B) (A - B) = A^2 - B^2$ then

Explanation

$$ \begin{aligned} A^2 &= \begin{vmatrix} 1 & -1 \\ 2 & 2 \end{vmatrix} \begin{vmatrix} 1 1 & -1 \\ 2 2 & 2 \end{vmatrix} \\ &= \begin{vmatrix} 1 1 & -3 \\ 6 & 2 \end{vmatrix} \\ &= -2 + 18 \\ \\ &= 16 \end{aligned} $$ $$ \begin{aligned} B^2 &= \begin{vmatrix} 1 & \alpha \\ \beta & 1 \end{vmatrix} \begin{vmatrix} 1 1 & \alpha \\ \beta \beta & 1 \end{vmatrix} \\ &= \begin{vmatrix} 1+\alpha\beta & 2\alpha \\ 2\beta & \alpha\beta+1 \end{vmatrix} \\ &= (1+\alpha\beta)^2 - 4\alpha\beta \\ &= (1-\alpha\beta)^2 \end{aligned} $$ $$ \begin{aligned} A + B &= \begin{vmatrix} 1 & -1 \\ 2 & 2 \end{vmatrix} + \begin{vmatrix} 1 1 & \alpha \\ \beta & 1 \end{vmatrix} \\ &= \begin{vmatrix} 2 2 & \alpha-1 \\ 2+\beta & 3 \end{vmatrix} \end{aligned} $$ $$ \begin{aligned} A - B &= \begin{vmatrix} 1 1 & -1 \\ 2 2 & 2 \end{vmatrix} - \begin{vmatrix} 1 1 & \alpha \\ \beta & 1 \end{vmatrix} \\ \\ &= \begin{vmatrix} 0 0 & -\alpha-1 \\ 2-\beta 2-\beta & 1 \end{vmatrix} \end{aligned} $$