Question

If \[ A = \begin{bmatrix} 1 & 3 \\ 2 & 2 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix}, \] then \[ B^{-1} A^{-1} = \]

• A. \( \begin{bmatrix} 2 & -3 \\ -7 & 11 \end{bmatrix} \)
• B. \( \begin{bmatrix} 2 & 3 \\ 7 & 11 \end{bmatrix} \)
• C. \( \begin{bmatrix} -2 & -3 \\ -7 & 11 \end{bmatrix} \)
• D. \( \begin{bmatrix} -2 & -3 \\ -7 & -11 \end{bmatrix} \)

Hint:

To find the inverse of a 2x2 matrix, use the formula \( A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \) and apply matrix multiplication.

Answer 1

The Correct Option is A

Step 1: Find the inverses of \( A \) and \( B \).
First, find the inverse of matrix \( A \) using the formula: \[ A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}, \] where \( \text{det}(A) = ad - bc \). Similarly, find the inverse of matrix \( B \).
Step 2: Multiply the inverses.
Next, multiply \( B^{-1} \) with \( A^{-1} \) to get the final result.
Step 3: Conclusion.
Thus, \( B^{-1} A^{-1} = \begin{bmatrix} 2 & -3 \\ -7 & 11 \end{bmatrix} \), corresponding to option (A).