Question

If the matrix $ A $ is such that $ A \begin{pmatrix} -1 & 2 \\ 3 & 1 \end{pmatrix} = \begin{pmatrix} -4 & 1 \\ 7 & 7 \end{pmatrix} \text{ then } A \text{ is equal to} $

• A. \( \begin{pmatrix} 1 & 2 \\ 1 & -3 \end{pmatrix} \)
• B. \( \begin{pmatrix} -1 & 2 \\ 1 & 3 \end{pmatrix} \)
• C. \( \begin{pmatrix} 1 & -2 \\ 1 & 3 \end{pmatrix} \)
• D. \( \begin{pmatrix} 1 & 2 \\ 3 & -1 \end{pmatrix} \)

Hint:

When solving matrix equations, use the inverse of the matrix to isolate the unknown matrix. Ensure to check the determinant of the matrix first, as the matrix must be invertible.

Answer 1

The Correct Option is D

We are given the equation: \[ A \begin{pmatrix} -1 & 2 3 & 1 \end{pmatrix} = \begin{pmatrix} -4 & 1 \\ 7 & 7 \end{pmatrix}. \] To find \( A \), we can multiply both sides by the inverse of the matrix \( \begin{pmatrix} -1 & 2 3 & 1 \end{pmatrix} \) from the right. 
Thus, we need to calculate the inverse of \( \begin{pmatrix} -1 & 2 3 & 1 \end{pmatrix} \). The determinant \( \text{det}(M) \) of the matrix \( M = \begin{pmatrix} -1 & 2 3 & 1 \end{pmatrix} \) is: \[ \text{det}(M) = (-1)(1) - (2)(3) = -1 - 6 = -7. \] The inverse of matrix \( M \) is given by: \[ M^{-1} = \frac{1}{\text{det}(M)} \begin{pmatrix} 1 & -2 -3 & -1 \end{pmatrix} = \frac{1}{-7} \begin{pmatrix} 1 & -2 -3 & -1 \end{pmatrix}. \] So, \[ M^{-1} = \begin{pmatrix} -\frac{1}{7} & \frac{2}{7} \frac{3}{7} & \frac{1}{7} \end{pmatrix}. \] Now, multiply both sides of the given equation by \( M^{-1} \) from the right: \[ A = \begin{pmatrix} -4 & 1 7 & 7 \end{pmatrix} \begin{pmatrix} -\frac{1}{7} & \frac{2}{7} \frac{3}{7} & \frac{1}{7} \end{pmatrix}. \] Perform the matrix multiplication: \[ A = \begin{pmatrix} -4 \cdot -\frac{1}{7} + 1 \cdot \frac{3}{7} & -4 \cdot \frac{2}{7} + 1 \cdot \frac{1}{7} 7 \cdot -\frac{1}{7} + 7 \cdot \frac{3}{7} & 7 \cdot \frac{2}{7} + 7 \cdot \frac{1}{7} \end{pmatrix}. \] Simplifying each element: \[ A = \begin{pmatrix} \frac{4}{7} + \frac{3}{7} & \\ -\frac{8}{7} + \frac{1}{7} -1 + 3 & 2 + 1 \end{pmatrix} = \begin{pmatrix} 1 & -1 \\ 2 & 3 \end{pmatrix}. \] 
Thus, the matrix \( A \) is: \[ A = \begin{pmatrix} 1 & 2 3 & -1 \end{pmatrix}. \] 
Hence, the correct answer is (D).