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Mathematics

If, x[23]+y[−11]=[105]x \begin{bmatrix} 2 \ 3 \end{bmatrix} + y \begin{bmatrix} -1 \ 1 \end{bmatrix} = \begin{bmatrix} 10 \ 5 \end{bmatrix}, then the value of xy is:

  1. 3

  2. -12

  3. -6

  4. 15

Matrices

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Answer

Solving for x and y:

⇒x[23]+y[−11]=[105]⇒[2x3x]+[−yy]=[105]⇒[2x−y3x+y]=[105]\Rightarrow x \begin{bmatrix} 2 \ 3 \end{bmatrix} + y \begin{bmatrix} -1 \ 1 \end{bmatrix} = \begin{bmatrix} 10 \ 5 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix} 2x \ 3x \end{bmatrix} + \begin{bmatrix} -y \ y \end{bmatrix} = \begin{bmatrix} 10 \ 5 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix} 2x - y \ 3x + y \end{bmatrix} = \begin{bmatrix} 10 \ 5 \end{bmatrix} \\[1em]

∴ 2x - y = 10 ….(1)

∴ 3x + y = 5 …..(2)

Adding equations (1) and (2), we get :

⇒ 2x - y + 3x + y = 10 + 5

⇒ 2x + 3x = 15

⇒ 5x = 15

⇒ x = 155\dfrac{15}{5}

⇒ x = 3.

Substituting value of x in 2x - y = 10, we get :

⇒ 2(3) - y = 10

⇒ 6 - y = 10

⇒ y = 6 - 10

⇒ y = -4.

∴ xy = (3)(-4) = -12.

Hence, option 2 is the correct option.

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