Question

If $X,Y$ are two matrices given by the equations $X+Y=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$ and $X-Y=\left[\begin{array}{cc}3& 2\\ -1& 0\end{array}\right],$ then $XY=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right].$ Find $a-b+c+d.$

Answer 1

Given,

$X+Y=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$ ....(i) and $X-Y=\left[\begin{array}{cc}3& 2\\ -1& 0\end{array}\right]$ ....(ii)

Adding (i) and (ii), we get

$2X=\left[\begin{array}{cc}4& 4\\ 2& 4\end{array}\right]\Rightarrow X=\left[\begin{array}{cc}2& 2\\ 1& 2\end{array}\right]$

By (i)$-$ (ii), we get

$2Y=\left[\begin{array}{cc}-2& 0\\ 4& 4\end{array}\right]\Rightarrow Y=\left[\begin{array}{cc}-1& 0\\ 2& 2\end{array}\right]$

$\therefore XY=\left[\begin{array}{cc}2& 2\\ 1& 2\end{array}\right]\left[\begin{array}{cc}-1& 0\\ 2& 2\end{array}\right]=\left[\begin{array}{cc}2& 4\\ 3& 4\end{array}\right]$
So, $a-b+c+d=5$