Question

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

Answer 1

Let $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

Given$A\left[\begin{array}{cc}1& -2\\ 1& 4\end{array}\right]=6I_2$

$⟹\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}1& -2\\ 1& 4\end{array}\right]=6\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

$⟹\left[\begin{array}{cc}a+b& -2a+4b\\ c+d& -2c+4d\end{array}\right]=\left[\begin{array}{cc}6& 0\\ 0& 6\end{array}\right]$

$⟹a+b=6$ ----(1)

$-2a+4b=0$ -----(2)

$c+d=0$ -----(3)

$-2c+4d=6$ -----(4)

$\left(1\right)\ast 2+\left(2\right)⟹6b=12⟹b=2$

$⟹a=6-2=4$

$\left(3\right)\ast 2+\left(4\right)⟹6d=6⟹d=1$

$⟹c=0-1=-1$

Therefore, $A=\left[\begin{array}{cc}4& 2\\ -1& 1\end{array}\right]$