Question

If $\left[\begin{array}{cc}x& 3x-y\\ 2x+z& 3y-\mathrm{\omega }\end{array}\right]=\left[\begin{array}{cc}3& 2\\ 4& 7\end{array}\right]$, find x, y, z, ω.

Answer 1

Since all the corresponding elements of a matrix are equal,
$$\begin{gathered} \left[\begin{array}{cc}x& 3x-y\\ 2x+z& 3y-w\end{array}\right]=\left[\begin{array}{cc}3& 2\\ 4& 7\end{array}\right] \\ x=3...\left(1\right) \\ 3x-y=2...\left(2\right) \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}x\mathrm{in}\mathrm{eq}.\left(2\right),\mathrm{we}\mathrm{get} \\ 3\left(3\right)-y=2 \\ \Rightarrow 9-y=2 \\ \Rightarrow -y=-7 \\ \Rightarrow y=7 \\ 2x+z=4...\left(3\right) \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}x\mathrm{in}\mathrm{eq}.\left(3\right),\mathrm{we}\mathrm{get} \\ 2\left(3\right)+z=4 \\ \Rightarrow 6+z=4 \\ \Rightarrow z=4-6 \\ \Rightarrow z=-2 \\ 3y-w=7...\left(4\right) \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}y\mathrm{in}\mathrm{eq}.\left(4\right),\mathrm{we}\mathrm{get} \\ 3\left(7\right)-w=7 \\ \Rightarrow 21-w=7 \\ \Rightarrow 21-7=w \\ \Rightarrow w=14 \\ \therefore x=3,y=7,z=-2\mathrm{and}w=14 \end{gathered}$$