Find x, y, z and t, if
Question:

Find xyz and t, if

(i) $3\left[\begin{array}{ll}x & y \\ z & t\end{array}\right]=\left[\begin{array}{rc}x & 6 \\ -1 & 2 t\end{array}\right]+\left[\begin{array}{cc}4 & x+y \\ z+t & 3\end{array}\right]$

(ii) $2\left[\begin{array}{cc}x & 5 \\ 7 & y-3\end{array}\right]+\left[\begin{array}{ll}3 & 4 \\ 1 & 2\end{array}\right]=\left[\begin{array}{rc}7 & 14 \\ 15 & 14\end{array}\right]$

Solution:

(i)

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

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

$\therefore 3 x=x+4$

$\Rightarrow 3 x-x=4$

$\Rightarrow 2 x=4$

$\Rightarrow x=2$

Also,

$3 y=6+x+y$

$\Rightarrow 3 y-y=6+x$

$\Rightarrow 2 y=6+x$

Putting the value of $x$ in eq. $(1)$, we get

$2 y=6+2$

$\Rightarrow 2 y=8$

$\Rightarrow y=4$

Now,

$3 t=2 t+3$

$\Rightarrow 3 t-2 t=3$

$\Rightarrow t=3$

$3 z=-1+z+t$

$\Rightarrow 3 z-z=-1+t$

$\Rightarrow 2 z=-1+t$                  $\ldots(2)$

Putting the value of $t$ in eq. (2), we get

$2 z=-1+3$

$\Rightarrow 2 z=2$

$\Rightarrow z=1$

$\therefore x=2, y=4, z=1$ and $t=3$

(ii)

$2\left[\begin{array}{cc}x & 5 \\ 7 & y-3\end{array}\right]+\left[\begin{array}{ll}3 & 4 \\ 1 & 2\end{array}\right]=\left[\begin{array}{cc}7 & 14 \\ 15 & 14\end{array}\right]$

$\Rightarrow\left[\begin{array}{cc}2 x & 10 \\ 14 & 2 y-6\end{array}\right]+\left[\begin{array}{ll}3 & 4 \\ 1 & 2\end{array}\right]=\left[\begin{array}{cc}7 & 14 \\ 15 & 14\end{array}\right]$

$\Rightarrow\left[\begin{array}{cc}2 x+3 & 10+4 \\ 14+1 & 2 y-6+2\end{array}\right]=\left[\begin{array}{cc}7 & 14 \\ 15 & 14\end{array}\right]$

$\Rightarrow\left[\begin{array}{cc}2 x+3 & 14 \\ 15 & 2 y-4\end{array}\right]=\left[\begin{array}{cc}7 & 14 \\ 15 & 14\end{array}\right]$

$\therefore 2 x+3=7$

$\Rightarrow 2 x=4$

$\Rightarrow x=2$

Also,

$2 y-4=14$

$\Rightarrow 2 y=18$

$\Rightarrow y=9$