Question

If inverse matrix of \( A = \begin{bmatrix} 2 & 3 \\ 1 & -4 \end{bmatrix} \) is \( A^{-1} = \begin{bmatrix} a & \frac{3}{11} \\ \frac{1}{11} & b \end{bmatrix} \), then a + b = _____

• A. \( \frac{2}{11} \)
• B. \( \frac{6}{11} \)
• C. \( -\frac{2}{11} \)
• D. \( -\frac{6}{11} \)

Hint:

Always use formula method for 2×2 inverse — fastest in exams.

Answer 1

The Correct Option is A

Concept: \[ A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \]
Step 1: Determinant. \[ |A| = (2)(-4) - (3)(1) = -8 - 3 = -11 \]
Step 2: Inverse. \[ A^{-1} = \frac{-1}{11} \begin{bmatrix} -4 & -3 \\ -1 & 2 \end{bmatrix} = \begin{bmatrix} \frac{4}{11} & \frac{3}{11} \\ \frac{1}{11} & -\frac{2}{11} \end{bmatrix} \]
Step 3: \[ a = \frac{4}{11}, \quad b = -\frac{2}{11} \] \[ a + b = \frac{2}{11} \]