If $M \times \begin{bmatrix} 1 & 1 \\ 0 & 2 \end{bmatrix} = \begin{bmatrix} 1 & 2 \end{bmatrix}$, find the order of matrix $M$ and the matrix $M$.
$M$ is a $1 \times 2$ matrix.
Let $M = \begin{bmatrix}
a & b
\end{bmatrix}$
Then, $\;$ $M \times \begin{bmatrix}
1 & 1 \\
0 & 2
\end{bmatrix} = \begin{bmatrix}
1 & 2
\end{bmatrix}$ $\implies$ $\begin{bmatrix}
a & b
\end{bmatrix} \times \begin{bmatrix}
1 & 1 \\
0 & 2
\end{bmatrix} = \begin{bmatrix}
1 & 2
\end{bmatrix}$
i.e. $\;$ $\begin{bmatrix}
a \times 1 + b \times 0 & a \times 1 + b \times 2
\end{bmatrix} = \begin{bmatrix}
1 & 2
\end{bmatrix}$
i.e. $\;$ $\begin{bmatrix}
a & a + 2b
\end{bmatrix} = \begin{bmatrix}
1 & 2
\end{bmatrix}$
When two matrices are equal, their corresponding elements are equal. Therefore, we have
$a = 1$
and $\;$ $a + 2b = 2$ $\implies$ $1 + 2b = 2$ $\implies$ $b = \dfrac{1}{2}$
$\therefore \;$ $M = \begin{bmatrix}
1 & \dfrac{1}{2}
\end{bmatrix}$