Matrices & Determinants
Matrix Multiplication
Grade 12
Question:
<p>If \(A = \begin{bmatrix} 1 & 2 & x \\ 3 & -1 & 2 \end{bmatrix}\) and \(B = \begin{bmatrix} y \\ x \\ 1 \end{bmatrix}\) be such that \(AB = \begin{bmatrix} 6 \\ 8 \end{bmatrix}\), then</p>
<p>\(y = 2x\)</p>
<p>\(y = -2x\)</p>
<p>\(y = x\)</p>
<p>\(y = x\)</p>
Step-by-Step Solution
Key Concept: Matrix multiplication AB requires matching dimensions (2×3)(3×1)=(2×1), then equate resulting equations to find unknowns x and y using the system of linear equations.
<p><strong>Step 1:</strong> Multiply matrices A(2×3) and B(3×1) to get a 2×1 matrix.</p><p>AB = $$\begin{bmatrix} 1 & 2 & x \\ 3 & -1 & 2 \end{bmatrix} \begin{bmatrix} y \\ x \\ 1 \end{bmatrix} = \begin{bmatrix} y + 2x + x \\ 3y - x + 2 \end{bmatrix} = \begin{bmatrix} y + 3x \\ 3y - x + 2 \end{bmatrix}$$</p><p><strong>Step 2:</strong> Equate to the given result matrix $$\begin{bmatrix} 6 \\ 8 \end{bmatrix}$$:</p><p>$$y + 3x = 6$$ ... (1)</p><p>$$3y - x + 2 = 8$$ ... (2)</p><p><strong>Step 3:</strong> From equation (2): $$3y - x = 6$$ ... (2')</p><p>From equation (1): $$y = 6 - 3x$$</p><p>Substitute into (2'): $$3(6 - 3x) - x = 6$$</p><p>$$18 - 9x - x = 6$$</p><p>$$-10x = -12$$</p><p>$$x = \frac{6}{5}$$</p><p>Then $$y = 6 - 3(\frac{6}{5}) = 6 - \frac{18}{5} = \frac{12}{5}$$</p><p>∴ Answer: A</p>
Correct Answer: A