Given:
$\begin{pmatrix}
4 & 2 \\
-1 & 1
\end{pmatrix} M = 6I$, where M is a matrix and I is an unit matrix of order $2 \times 2$.
State the order of matrix M.
Find the matrix M.
Order of matrix $M = 2 \times 2$
Let $M=$
$\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}$, $I=$
$\begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix} =$ unit matrix
$\therefore$
$\begin{pmatrix}
4 & 2 \\
-1 & 1
\end{pmatrix} M = 6 I \implies$
$\begin{pmatrix}
4 & 2 \\
-1 & 1
\end{pmatrix}$
$\begin{pmatrix}
a & b \\
c & d
\end{pmatrix} = 6$
$\begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}$
i.e.
$\begin{pmatrix}
4a + 2c & 4b+2d \\
-a+c & -b+d
\end{pmatrix} =$
$\begin{pmatrix}
6 & 0 \\
0 & 6
\end{pmatrix}$
i.e. $4a + 2c = 6$ $\implies$ $2a + c = 3 \cdots (1)$
$-a+c = 0$ $\implies$ $a=c$ $\cdots$ (2)
$4b+2d=0$ $\implies$ $d = -2b$ $\cdots$ (3)
$-b+d=6$ $\cdots$ (4)
From equations (1) and (2) we have
$2c + c = 3$ $\implies$ $c=1$
Therefore, from equation (2), $a = 1$
From equations (3) and (4) we have
$-b-2b = 6$ $\implies$ $b=-2$
Therefore, from equation (3), $d=4$
$\therefore$ $M =$
$\begin{pmatrix}
1 & -2 \\
1 & 4
\end{pmatrix}$