Question

If \( A = \begin{bmatrix} 4 & 5 2 & 1 \end{bmatrix} \), find \( A^{-1} \).

• A. \( \begin{bmatrix} 1/6 & -5/6 -1/3 & 2/3 \end{bmatrix} \)
• B. \( \begin{bmatrix} -1/6 & 5/6 1/3 & -2/3 \end{bmatrix} \)
• C. \( \begin{bmatrix} -1 & 5 2 & -4 \end{bmatrix} \)
• D. \( \begin{bmatrix} 4 & -5 -2 & 1 \end{bmatrix} \)

Hint:

For \(2 \times 2\) matrices, always use the shortcut: Swap diagonal elements, change signs of off-diagonal elements, then divide by determinant.

Answer 1

The Correct Option is B

To find the inverse of a 2x2 matrix \( A \), we can use the formula for the inverse of a matrix:

\( A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \) 

where \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \).

The given matrix is:

\( A = \begin{bmatrix} 4 & 5 \\ 2 & 1 \end{bmatrix} \)

1. First, calculate the determinant of \( A \), denoted as \( \text{det}(A) \):
2. Now, apply the formula for the inverse of the matrix:
3. Simplify the matrix:

Thus, the inverse of matrix \( A \) is:

\( A^{-1} = \begin{bmatrix} -1/6 & 5/6 \\ 1/3 & -2/3 \end{bmatrix} \)

Therefore, the correct answer is:

\( \begin{bmatrix} -1/6 & 5/6 \\ 1/3 & -2/3 \end{bmatrix} \)

Option 2 is the correct answer, as per the explained calculation.