Question

If \( A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \), then \( A^{-1} \) is:

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

Hint:

Remember the inverse formula for a 2x2 matrix. It's a fundamental operation. Pay close attention to signs when changing off-diagonal elements and when multiplying by the inverse of the determinant.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:

The question asks to find the inverse of a given 2×2 matrix \(A\).

Step 2: Key Formula or Approach:

For a 2×2 matrix \[ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \] its inverse \(A^{-1}\) is given by: \[ A^{-1} = \frac{1}{\det(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \] where the determinant of \(A\) is: \[ \det(A) = ad - bc \]

Step 3: Detailed Explanation:

Given matrix: \[ A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \]

Here, \(a = 1,\; b = 2,\; c = 3,\; d = 4\).

1. Calculate the determinant of \(A\):

\[ \det(A) = ad - bc \] \[ = (1)(4) - (2)(3) \] \[ = 4 - 6 = -2 \]

2. Find the adjoint of \(A\):

Swap the diagonal elements and change the signs of the off-diagonal elements: \[ \text{adj}(A) = \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix} \]

3. Calculate the inverse of \(A\):

\[ A^{-1} = \frac{1}{-2} \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix} \]

Multiplying each element: \[ A^{-1} = \begin{pmatrix} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{pmatrix} \]

Step 4: Final Answer:

Therefore, the inverse matrix is: \[ A^{-1} = \begin{pmatrix} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{pmatrix} \]