Find x, y, z 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) $2\left[\begin{array}{cc}x & 5 \\ 7 & y-3\end{array}\right]+\left[\begin{array}{ll}3 & 4 \\ 1 & 2\end{array}\right]=\left[\begin{array}{rc}7 & 14 \\ 15 & 14\end{array}\right]$
(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,
$2 y-4=14$
$\Rightarrow 2 y=18$
$\Rightarrow y=9$
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