Question

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

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

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right) \\ 3\left[\begin{array}{cc}x& y\\ z& t\end{array}\right]=\left[\begin{array}{cc}x& 6\\ -1& 2t\end{array}\right]+\left[\begin{array}{cc}4& x+y\\ z+t& 3\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}3x& 3y\\ 3z& 3t\end{array}\right]=\left[\begin{array}{cc}x+4& 6+x+y\\ -1+z+t& 2t+3\end{array}\right] \\ \therefore 3x=x+4 \\ \Rightarrow 3x-x=4 \\ \Rightarrow 2x=4 \\ \Rightarrow x=2 \\ \mathrm{Also}, \\ 3y=6+x+y \\ \Rightarrow 3y-y=6+x \\ \Rightarrow 2y=6+x...\left(1\right) \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}x\mathrm{in}\mathrm{eq}.\left(1\right),\mathrm{we}\mathrm{get} \\ 2y=6+2 \\ \Rightarrow 2y=8 \\ \Rightarrow y=4 \\ \mathrm{Now}, \\ 3t=2t+3 \\ \Rightarrow 3t-2t=3 \\ \Rightarrow t=3 \\ 3z=-1+z+t \\ \Rightarrow 3z-z=-1+t \\ \Rightarrow 2z=-1+t...\left(2\right) \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}t\mathrm{in}\mathrm{eq}.\left(2\right),\mathrm{we}\mathrm{get} \\ 2z=-1+3 \\ \Rightarrow 2z=2 \\ \Rightarrow z=1 \\ \therefore x=2,y=4,z=1\mathrm{and}t=3 \end{gathered}$$


$$\begin{gathered} \left(\mathrm{ii}\right) \\ 2\left[\begin{array}{cc}x& 5\\ 7& y-3\end{array}\right]+\left[\begin{array}{cc}3& 4\\ 1& 2\end{array}\right]=\left[\begin{array}{cc}7& 14\\ 15& 14\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}2x& 10\\ 14& 2y-6\end{array}\right]+\left[\begin{array}{cc}3& 4\\ 1& 2\end{array}\right]=\left[\begin{array}{cc}7& 14\\ 15& 14\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}2x+3& 10+4\\ 14+1& 2y-6+2\end{array}\right]=\left[\begin{array}{cc}7& 14\\ 15& 14\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}2x+3& 14\\ 15& 2y-4\end{array}\right]=\left[\begin{array}{cc}7& 14\\ 15& 14\end{array}\right] \\ \therefore 2x+3=7 \\ \Rightarrow 2x=4 \\ \Rightarrow x=2 \\ \mathrm{Also}, \\ 2y-4=14 \\ \Rightarrow 2y=18 \\ \Rightarrow y=9 \end{gathered}$$