If m × $\begin{bmatrix$}[r] 2 \ 3 \end{bmatrix} + n\begin{bmatrix}[r] -1 \ 1 \end{bmatrix} = \begin{bmatrix}[r] 10 \ 5 \end{bmatrix}, then the values of m and n are :

1. m = -3, n = -4

2. m = 3, n = -4

3. m = -3, n = 4

4. m = 3, n = 4

Solving,

$\Rightarrow m \times \begin{bmatrix$}[r] 2 \ 3 \end{bmatrix} + n\begin{bmatrix}[r] -1 \ 1 \end{bmatrix} = \begin{bmatrix}[r] 10 \ 5 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 2m \ 3m \end{bmatrix} + \begin{bmatrix}[r] -n \ n \end{bmatrix} = \begin{bmatrix}[r] 10 \ 5 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 2m - n \ 3m + n \end{bmatrix} = \begin{bmatrix}[r] 10 \ 5 \end{bmatrix}.

From above,

1st equation :

⇒ 2m - n = 10

⇒ n = 2m - 10 ………(1)

2nd equation :

⇒ 3m + n = 5

⇒ n = 5 - 3m ……..(2)

From (1) and (2), we get :

⇒ 2m - 10 = 5 - 3m

⇒ 2m + 3m = 5 + 10

⇒ 5m = 15

⇒ m = $\dfrac{15}{5}$ = 3.

Substituting value of m in equation (1), we get :

⇒ n = 2(3) - 10 = 6 - 10 = -4.

∴ m = 3 and n = -4.

Hence, Option 2 is the correct option.