If $x\begin{bmatrix$}[r] 2 \ 3 \end{bmatrix} + y\begin{bmatrix}[r] -1 \ 0 \end{bmatrix} = \begin{bmatrix}[r] 10 \ 6 \end{bmatrix}, then the values of x and y are

1. x = 2, y = 6
2. x = 2, y = -6
3. x = 3, y = -4
4. x = 3, y = -6

Given,

$x\begin{bmatrix$}[r] 2 \ 3 \end{bmatrix} + y\begin{bmatrix}[r] -1 \ 0 \end{bmatrix} = \begin{bmatrix}[r] 10 \ 6 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 2x \ 3x \end{bmatrix} + \begin{bmatrix}[r] -y \ 0 \end{bmatrix} = \begin{bmatrix}[r] 10 \ 6 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 2x - y \ 3x \end{bmatrix} = \begin{bmatrix}[r] 10 \ 6 \end{bmatrix} \\[1em]

By definition of equality of matrices we get,

⇒ 2x - y = 10          (…Eq 1)

⇒ 3x = 6 or x = 2.

Putting value of x in Eq 1

⇒ 2x - y = 10
⇒ 2(2) - y = 10
⇒ 4 - y = 10
⇒ y = 4 - 10
⇒ y = -6

∴ x = 2 and y = -6.

∴ Option 2 is the correct option.