Find the adjoint of the matrix \( A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{bmatrix} \)
Answer
The adjoint of A is the transpose of the cofactor matrix C. Hence, \[ adj(A) = C^T = \begin{bmatrix} 24 & -30 & 20 \\ 0 & 6 & -5 \\ 0 & 0 & 1 \end{bmatrix} \]
Step 1 ：Calculate the cofactor matrix of A. The cofactor of an element \( a_{ij} \) is given by \( (-1)^{i+j} \) times the determinant of the matrix obtained by removing the i-th row and j-th column. In this case, the cofactor matrix C of A is \[ C = \begin{bmatrix} 24 & 0 & 0 \\ -30 & 6 & 0 \\ 20 & -5 & 1 \end{bmatrix} \]
Step 2 ：The adjoint of A is the transpose of the cofactor matrix C. Hence, \[ adj(A) = C^T = \begin{bmatrix} 24 & -30 & 20 \\ 0 & 6 & -5 \\ 0 & 0 & 1 \end{bmatrix} \]
