Question

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

Answer 1

Given: $3\left[\begin{array}{cc}x& y\\ z& w\end{array}\right]=\left[\begin{array}{cc}x& 6\\ -1& 2w\end{array}\right]+\left[\begin{array}{cc}4& x+y\\ z+w& 3\end{array}\right]$
$\Rightarrow \left[\begin{array}{cc}3x& 3y\\ 3z& 3w\end{array}\right]=\left[\begin{array}{cc}x+4& 6+x+y\\ -1+z+w& 2w+3\end{array}\right]$
Comparing, we get
$3x=x+4\cdots \left(i\right)$
$3y=6+x+y\cdots \left(ii\right)$
$3z=-1+z+w\cdots \left(iii\right)$
$3w=2w+3\cdots \left(iv\right)$
Solving equation $\left(i\right)$
$\Rightarrow 3x=x+4$
$\Rightarrow 3x-x=4\Rightarrow 2x=4$
$\Rightarrow x=\frac{4}{2}=2$
Solving equation $\left(ii\right)$
$\Rightarrow 3y=6+x+y$
$\Rightarrow 3y-y=6+x$
$\Rightarrow 2y=6+x$
Putting $x=2$
$\Rightarrow 2y=6+2$
$\Rightarrow 2y=8\Rightarrow y=4$
Solving equation $\left(iv\right)$
$\Rightarrow 3w=2w+3$
$\Rightarrow 3w-2w=3$
$\Rightarrow w=3$
Solving equation $\left(iii\right)$
$\Rightarrow 3z=-1+z+w$
$\Rightarrow 3z-z=-1+w$
$\Rightarrow 2z=-1+w$
Putting $w=3$
$\Rightarrow 2z=-1+3$
$\Rightarrow 2z=2$
$\Rightarrow z=\frac{2}{2}=1$
$\therefore x=2,y=4,w=3$ and $z=1$