<p>If \(AB = \dfrac{6}{8}\) and \(\begin{bmatrix}1 & 2 & x \\ 3 & -1 & 2\end{bmatrix}_{2\times3} \begin{bmatrix} y \\ x \\ 1 \end{bmatrix}_{3\times1} = \begin{bmatrix}6\\8\end{bmatrix}\), find \(y\).</p>
Step-by-Step Solution
Key Concept: Matrix multiplication follows row-by-column dot product rule; the first row of matrix A multiplied by the column vector gives the first component of the result, and the second row gives the second component. This creates a system of two linear equations in two unknowns (x and y).
<p><strong>Step 1:</strong> Perform matrix multiplication on the left side:</p><p>$$\begin{bmatrix}1 & 2 & x \\ 3 & -1 & 2\end{bmatrix} \begin{bmatrix} y \\ x \\ 1 \end{bmatrix} = \begin{bmatrix} 1(y) + 2(x) + x(1) \\ 3(y) + (-1)(x) + 2(1) \end{bmatrix} = \begin{bmatrix} y + 3x \\ 3y - x + 2 \end{bmatrix}$$</p><p><strong>Step 2:</strong> Set this equal to the given result:</p><p>$$\begin{bmatrix} y + 3x \\ 3y - x + 2 \end{bmatrix} = \begin{bmatrix}6\\8\end{bmatrix}$$</p><p><strong>Step 3:</strong> This gives us the system of equations:</p><p>$$y + 3x = 6 \quad \text{...(1)}$$</p><p>$$3y - x + 2 = 8 \quad \text{...(2)}$$</p><p><strong>Step 4:</strong> Simplify equation (2):</p><p>$$3y - x = 6 \quad \text{...(2')}$$</p><p><strong>Step 5:</strong> From equation (1): $y = 6 - 3x$</p><p><strong>Step 6:</strong> Substitute into equation (2'):</p><p>$$3(6 - 3x) - x = 6$$</p><p>$$18 - 9x - x = 6$$</p><p>$$18 - 10x = 6$$</p><p>$$10x = 12$$</p><p>$$x = \frac{6}{5}$$</p><p><strong>Step 7:</strong> Substitute back into equation (1):</p><p>$$y + 3\left(\frac{6}{5}\right) = 6$$</p><p>$$y + \frac{18}{5} = 6$$</p><p>$$y = 6 - \frac{18}{5} = \frac{30 - 18}{5} = \frac{12}{5}$$</p><p>∴ Answer: <strong>y = 12/5</strong></p>
Correct Answer: A