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

To find the inverse of a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \), we use the formula:

\(A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}\)

Let's apply this formula to the given matrix \( A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \).

1. First, find the determinant of \( A \).

\(\text{det}(A) = (1)(4) - (2)(3) = 4 - 6 = -2\)

1. Now, apply the inverse formula:

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

Calculate each element in the inverse matrix:

• \(A^{-1} = \begin{pmatrix} \frac{4}{-2} & \frac{-2}{-2} \\ \frac{-3}{-2} & \frac{1}{-2} \end{pmatrix}\)
• Simplify the elements:
• \(A^{-1} = \begin{pmatrix} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{pmatrix}\)

So, the inverse of \( A \) is:

\( A^{-1} = \begin{pmatrix} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{pmatrix} \)

Therefore, the correct option is:

• \( \begin{pmatrix} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{pmatrix} \)