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

$3\left|\begin{array}{cc}x& y\\ z& w\end{array}\right|=\left|\begin{array}{cc}x& 6\\ -1& 2w\end{array}\right|+\left|\begin{array}{cc}4& x+y\\ z+w& 3\end{array}\right|$

As we know that addition of two matrix is preformed by adding their corresponding elements.

$\therefore \left|\begin{array}{cc}3x& 3y\\ 3z& 3w\end{array}\right|=\left|\begin{array}{cc}x+4& 6+x+y\\ -1+z+w& 2w+3\end{array}\right|$

As we also know that two matrices are equal if their corresponding elements are equal.

Therefore

• $3x=x+4\Rightarrow x=2$
  
• $3y=6+x+y\Rightarrow y=4$
• $3w=2w+3\Rightarrow w=3$
• $3z=-1+z+w\Rightarrow z=1$
• Hence the values of $x,y,z$ and $w$ are $2,4,1$ and $3$ respectively.