Question

If \( A = \begin{bmatrix} 3 & -5 \\ -1 & 2 \end{bmatrix} \), then \( \text{adjoint } A = \):

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

Hint:

For a 2×2 matrix \( A = \begin{bmatrix} a & b
c & d \end{bmatrix} \), the adjoint is \( \begin{bmatrix} d & -b
-c & a \end{bmatrix} \).

Answer 1

The Correct Option is A

Given \( A = \begin{bmatrix} 3 & -5 \\ -1 & 2 \end{bmatrix} \), we find the adjoint of a 2×2 matrix by swapping the diagonal elements and changing the sign of the off-diagonal elements: \[ \text{If } A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, \quad \text{then } \operatorname{adj}(A) = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \] Applying this: \[ \operatorname{adj}(A) = \begin{bmatrix} 2 & 5 \\ 1 & 3 \end{bmatrix} \]