Question

If \( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \), verify that \( A (\text{adj } A) = (\text{adj } A) A = |A| I \).

Hint:

For a square matrix, \( A (\text{adj } A) = |A| I \); compute adjoint using cofactors.

Answer 1

Compute determinant: \( |A| = 1 \cdot 4 - 2 \cdot 3 = 4 - 6 = -2 \).
Adjoint: Cofactors of \( A \): - \( C_{11} = 4 \), \( C_{12} = -3 \), \( C_{21} = -2 \), \( C_{22} = 1 \).
\[ \text{adj } A = \begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix}. \] Verify \( A (\text{adj } A) \): \[ A (\text{adj } A) = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix} = \begin{bmatrix} 1 \cdot 4 + 2 \cdot (-3) & 1 \cdot (-2) + 2 \cdot 1
3 \cdot 4 + 4 \cdot (-3) & 3 \cdot (-2) + 4 \cdot 1 \end{bmatrix} = \begin{bmatrix} -2 & 0
0 & -2 \end{bmatrix} = (-2) I. \] Verify \( (\text{adj } A) A \): \[ (\text{adj } A) A = \begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 4 \cdot 1 + (-2) \cdot 3 & 4 \cdot 2 + (-2) \cdot 4 \\ (-3) \cdot 1 + 1 \cdot 3 & (-3) \cdot 2 + 1 \cdot 4 \end{bmatrix} = \begin{bmatrix} -2 & 0 \\ 0 & -2 \end{bmatrix} = (-2) I. \] Since \( |A| = -2 \), \( A (\text{adj } A) = (\text{adj } A) A = |A| I \).
Answer: Property verified.