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

Step 1: Understanding the Concept:
For a $2 \times 2$ matrix $A = \begin{pmatrix} a' & b'
c' & d' \end{pmatrix}$, the inverse is $A^{-1} = \frac{1}{|A|} \begin{pmatrix} d' & -b'
-c' & a' \end{pmatrix}$.
Step 2: Formula Application:
$|A| = (2)(-4) - (3)(1) = -8 - 3 = -11$. $A^{-1} = \frac{1}{-11} \begin{pmatrix} -4 & -3
-1 & 2 \end{pmatrix} = \begin{pmatrix} 4/11 & 3/11
1/11 & -2/11 \end{pmatrix}$
Step 3: Explanation:
Comparing this with the given $A^{-1}$: $a = 4/11$ and $b = -2/11$. $a + b = 4/11 + (-2/11) = 2/11$.
Step 4: Final Answer:
The sum $a + b$ is $2/11$.