Question

If $3\left[\begin{array}{cc}x& y\\ z& w\end{array}\right]=\left[\begin{array}{cc}x& 6\\ -1& 2w\end{array}\right]+\left[\begin{array}{cc}4& x+y\\ x+w& 3\end{array}\right]$, find the values of x, y, z and w.

Answer 1

Given, $3\left[\begin{array}{cc}x& y\\ x& w\end{array}\right]=\left[\begin{array}{cc}x& 6\\ -1& 2w\end{array}\right]+\left[\begin{array}{cc}4& x+y\\ x+w& 3\end{array}\right]$
$\Rightarrow \left[\begin{array}{cc}3x& 3y\\ 3z& 3w\end{array}\right]=\left[\begin{array}{cc}x+4& 6+x+y\\ -1+z+w& 2w+3\end{array}\right]$

By definition of equality of matrix as the given matrices are equal, their corresponding elements are equal. Comparing the corresponding elements, we get
$3x=x+4\Rightarrow 2x=4\Rightarrow x=2$
and $3y=6+x+y\Rightarrow 2y=6+x\Rightarrow y=\frac{6+x}{2}$
Putting the value of x, we get
$y=\frac{6+2}{2}=\frac{8}{2}=4$
Now, 2z=-1 +z+w, z=-1 +w
$z=\frac{-1+w}{2}.......\left(i\right)$
Now, $3w=2w+3\Rightarrow w=3$
Putting the value of w in Eq. (i), we get
$z=\frac{-1+3}{2}=\frac{2}{2}=1$
Hence, the values of x,y, z and w are 2,4,1 and 3.