Question

If \( A = \begin{bmatrix} 1 & 1 \\ 2 & 1 \end{bmatrix}, B = \begin{bmatrix} 4 & 3 \\ 1 & 1 \end{bmatrix} \), then \( (A + B)^{-1x} = \)

• A. \( \frac{1}{7} \begin{bmatrix} 3 & 4 \\ 2 & 5 \end{bmatrix} \)
• B. \( 7 \begin{bmatrix} 3 & 4 \\ 2 & 5 \end{bmatrix} \)
• C. \( \frac{1}{7} \begin{bmatrix} 3 & -2 \\ -4 & 5 \end{bmatrix} \)
• D. \( 7 \begin{bmatrix} 3 & -2 \\ -4 & 5 \end{bmatrix} \)

Hint:

To find the inverse of a 2x2 matrix, use the formula for the inverse and ensure you calculate the determinant correctly.

Answer 1

The Correct Option is C

Step 1: Add the matrices.
First, find \( A + B \): \[ A + B = \begin{bmatrix} 1 & 1 \\2 & 1 \end{bmatrix} + \begin{bmatrix} 4 & 3 \\ 1 & 1 \end{bmatrix} = \begin{bmatrix} 5 & 4 \\ 3 & 2 \end{bmatrix} \] Step 2: Find the inverse.
Next, calculate the inverse of \( A + B \). The inverse of a 2x2 matrix \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) is given by: \[ \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \] For \( A + B = \begin{bmatrix} 5 & 4 \\ 3 & 2 \end{bmatrix} \), the determinant is \( 5(2) - 4(3) = -2 \). Therefore, the inverse is: \[ \frac{1}{-2} \begin{bmatrix} 2 & -4 \\ -3 & 5 \end{bmatrix} = \frac{1}{7} \begin{bmatrix} 3 & -2 \\ -4 & 5 \end{bmatrix} \] Step 3: Conclusion.
The correct answer is (C) \( \frac{1}{7} \begin{bmatrix} 3 & -2 \\ -4 & 5 \end{bmatrix} \).