Question

Find a 2 × 2 matrix A such that
$A\left[\begin{array}{cc}1& -2\\ 1& 4\end{array}\right]=6I_2$

Answer 1

Let A = $\left[\begin{array}{cc}w& x\\ y& z\end{array}\right]$

Now,

$$\begin{gathered} \left[\begin{array}{cc}w& x\\ y& z\end{array}\right]\left[\begin{array}{cc}1& -2\\ 1& 4\end{array}\right]=6I_2 \\ \Rightarrow \left[\begin{array}{cc}w+x& -2w+4x\\ y+z& -2y+4z\end{array}\right]=6\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}w+x& -2w+4x\\ y+z& -2y+4z\end{array}\right]=\left[\begin{array}{cc}6& 0\\ 0& 6\end{array}\right] \\ \mathrm{The}\mathrm{corresponding}\mathrm{elements}\mathrm{of}\mathrm{two}\mathrm{equal}\mathrm{matrices}\mathrm{are}\mathrm{equal}. \\ \therefore w+x=6 \\ \Rightarrow w=6-x...\left(1\right) \\ -2w+4x=0...\left(2\right) \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}w\mathrm{in}eq.\left(2\right),\mathrm{we}\mathrm{get} \\ -2\left(6-x\right)+4x=0 \\ \Rightarrow -12+2x+4x=0 \\ \Rightarrow -12+6x=0 \\ \Rightarrow 6x=12 \\ \Rightarrow x=2 \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}x\mathrm{in}\mathrm{eq}.\left(1\right),\mathrm{we}\mathrm{get} \\ w=6-2 \\ \Rightarrow w=4 \\ \mathrm{Now}, \\ y+z=0 \\ \Rightarrow y=-z...\left(3\right) \\ -2y+4z=6...\left(4\right) \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}y\mathrm{in}\mathrm{eq}.\left(4\right),\mathrm{we}\mathrm{get} \\ -2\left(-z\right)+4z=6 \\ \Rightarrow 2z+4z=6 \\ \Rightarrow 6z=6 \\ \Rightarrow z=1 \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}z\mathrm{in}\mathrm{eq}.\left(3\right),\mathrm{we}\mathrm{get} \\ y=-1 \\ \therefore A=\left[\begin{array}{cc}4& 2\\ -1& 1\end{array}\right] \end{gathered}$$