Question

If \[ A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \] verify that \[ A(\text{adj}A) = (\text{adj}A)A = |A|I \]

Hint:

To verify matrix identities involving the adjoint of a matrix, compute the adjoint and perform the matrix multiplications.

Answer 1

Step 1: First, compute the determinant \( |A| \) of matrix \( A \): \[ |A| = \begin{vmatrix} 1 & 2 \\ 3 & 4 \end{vmatrix} = (1)(4) - (2)(3) = 4 - 6 = -2 \] Step 2: Next, find the adjoint of matrix \( A \), which is the transpose of the cofactor matrix. The cofactor matrix is: \[ \text{Cofactor matrix of } A = \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix} \] The adjoint of \( A \) is the transpose of the cofactor matrix: \[ \text{adj}A = \begin{pmatrix} 4 & -3 \\ -2 & 1 \end{pmatrix} \] Step 3: Now, compute \( A \cdot (\text{adj}A) \): \[ A \cdot (\text{adj}A) = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \cdot \begin{pmatrix} 4 & -3 \\ -2 & 1 \end{pmatrix} \] \[ = \begin{pmatrix} (1)(4) + (2)(-2) & (1)(-3) + (2)(1) \\ (3)(4) + (4)(-2) & (3)(-3) + (4)(1) \end{pmatrix} \] \[ = \begin{pmatrix} 4 - 4 & -3 + 2 \\ 12 - 8 & -9 + 4 \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 4 & -5 \end{pmatrix} \] Step 4: Finally, compute \( |A| \cdot I \), where \( I \) is the identity matrix: \[ |A| \cdot I = (-2) \cdot \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} -2 & 0 \\ 0 & -2 \end{pmatrix} \] Step 5: As we can see, \( A \cdot (\text{adj}A) = (\text{adj}A) \cdot A = |A| \cdot I \), which verifies the result. Thus, we have verified that: \[ A(\text{adj}A) = (\text{adj}A)A = |A|I \]