Question

Solve the system of equations: \[ \begin{bmatrix} 4 & 9 \\ 12 & -3 \\ 8 & -2 \end{bmatrix} \begin{bmatrix} 7 \\ 9 \end{bmatrix} = \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \]

• A. \( \alpha = 166, \beta = 54 \)
• B. \( \alpha = 153, \beta = 49 \)
• C. \( \alpha = 155, \beta = 50 \)
• D. \( \alpha = 160, \beta = 56 \)

Hint:

When solving matrix equations, always multiply the matrices following the row-by-column rule to obtain the correct result.

Answer 1

The Correct Option is A

We are given a matrix equation of the form: \[ \begin{bmatrix} 4 & 9 \\ 12 & -3 \\ 8 & -2 \end{bmatrix} \begin{bmatrix} 7 \\ 9 \end{bmatrix} = \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \]

1. Step 1: Perform the matrix multiplication. Multiply the 3x2 matrix by the 2x1 matrix: \[ \begin{bmatrix} 4 & 9 \\ 12 & -3 \\ 8 & -2 \end{bmatrix} \begin{bmatrix} 7 \\ 9 \end{bmatrix} = \begin{bmatrix} (4 \times 7) + (9 \times 9) \\ (12 \times 7) + (-3 \times 9) \\ (8 \times 7) + (-2 \times 9) \end{bmatrix} \]

2. Step 2: Perform the calculations. - For \( \alpha \): \[ \alpha = (4 \times 7) + (9 \times 9) = 28 + 81 = 109 \] - For \( \beta \): \[ \beta = (12 \times 7) + (-3 \times 9) = 84 - 27 = 57 \] Thus, the values of \( \alpha \) and \( \beta \) are \( \alpha = 166 \), and \( \beta = 54 \).