Question

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

Answer 1

$X+Y=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]⟶\left(1\right)$

$X-Y=\left[\begin{array}{cc}3& 2\\ -1& 0\end{array}\right]⟶\left(2\right)$

Adding equation $\left(1\right)$ & $\left(2\right),$ we get

$\Rightarrow 2X=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]+\left[\begin{array}{cc}3& 2\\ -1& 0\end{array}\right]$

$\Rightarrow 2X=\left[\begin{array}{cc}1+3& 2+2\\ 3-1& 4+0\end{array}\right]$

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

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

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

Again, subtracting equation $\left(1\right)$ & $\left(2\right)$, we get

$\Rightarrow 2Y=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]-\left[\begin{array}{cc}3& 2\\ -1& 0\end{array}\right]$

$\Rightarrow 2Y=\left[\begin{array}{cc}1-3& 2-2\\ 3+1& 4-0\end{array}\right]$

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

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

Now, $XY=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

$\Rightarrow \left[\begin{array}{cc}2& 2\\ 1& 2\end{array}\right]\left[\begin{array}{cc}-1& 0\\ 2& 2\end{array}\right]=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

$\Rightarrow \left[\begin{array}{cc}-2+4& 0+4\\ -1+4& 0+4\end{array}\right]=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

$\Rightarrow \left[\begin{array}{cc}2& 4\\ 3& 4\end{array}\right]=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

Comparing both sides, we get

$a=2$

$b=4$

$c=3$

and $d=4$

Hence, the answer is $a=2,b=4,c=3$ and $d=4.$