Question

If $ A = \begin{bmatrix} 2 & 2 \\3 & 4 \end{bmatrix}, \quad \text{then} \quad A^{-1} \text{ equals to} $

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

Hint:

For any 2x2 matrix, the inverse can be found using the formula \(\frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \\-c & a \end{bmatrix}\), provided the determinant is not zero.

Answer 1

The Correct Option is B

To find \(A^{-1}\), we use the formula for the inverse of a 2x2 matrix: \[ A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b -c & a \end{bmatrix} \] where \(a = 2, b = 2, c = 3, d = 4\). The determinant is: \[ \text{det}(A) = ad - bc = 2(4) - 2(3) = 8 - 6 = 2 \] Now, using the formula for the inverse: \[ A^{-1} = \frac{1}{2} \begin{bmatrix} 4 & -2 -3 & 2 \end{bmatrix} = \begin{bmatrix} 2 & -1 \\-3/2 & 1 \end{bmatrix} \]