Let M $\times \begin{bmatrix$}[r] 1 & 1 \ 0 & 2 \end{bmatrix} = \begin{bmatrix}[r] 1 & 2 \end{bmatrix} where M is a matrix.

(i) State the order of the matrix M.

(ii) Find the matrix M.

(i)

Since,

$\text{M} \times \begin{bmatrix$}[r] 1 & 1 \ 0 & 2 \end{bmatrix} = \begin{bmatrix}[r] 1 & 2 \end{bmatrix} \\[1em] \Rightarrow \text{M} \times \begin{bmatrix}[r] 1 & 1 \ 0 & 2 \end{bmatrix} \text{is a } 1 \times 2 \text{ matrix, but} \begin{bmatrix}[r] 1 & 1 \ 0 & 2 \end{bmatrix} \text{ is a } 2 \times 2 \text{ matrix}. \\[1em] \Rightarrow \text{M is a } 1 \times 2 \text{ matrix}.

The order of matrix M is 1 × 2.

(ii)

$\text{Let M =} \begin{bmatrix$}[r] x & y \end{bmatrix} \\[1em] \text{Given, } \text{M} \times \begin{bmatrix}[r] 1 & 1 \ 0 & 2 \end{bmatrix} = \begin{bmatrix}[r] 1 & 2 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] x & y \end{bmatrix} \times \begin{bmatrix}[r] 1 & 1 \ 0 & 2 \end{bmatrix} = \begin{bmatrix}[r] 1 & 2 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] x \times 1 + y \times 0 & x \times 1 + y \times 2 \ \end{bmatrix} = \begin{bmatrix}[r] 1 & 2 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] x & x + 2y \ \end{bmatrix} = \begin{bmatrix}[r] 1 & 2 \end{bmatrix} \\[1em]

By definition of equality of matrices we get,

x = 1                (…Eq 1)
x + 2y = 2        (…Eq 2)

Putting value of x from Eq 1 in Eq 2,

⇒ x + 2y = 2
⇒ 1 + 2y = 2
⇒ 2y = 2 - 1
⇒ 2y = 1
⇒ y = $\dfrac{1}{2}$

$\text{Since, M }= \begin{bmatrix$}[r] x & y \ \end{bmatrix} \\[1em] \therefore \text{M} = \begin{bmatrix}[r] 1 & \dfrac{1}{2} \ \end{bmatrix}.

Hence, the matrix M = $\begin{bmatrix$}[r] 1 & \dfrac{1}{2} \end{bmatrix}.