Find the adjoint of the matrix \( A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{bmatrix} \).

Step 1 ：The adjoint of a matrix is the transpose of the cofactor matrix. Therefore, we first need to find the cofactor matrix of \( A \).

Step 2 ：The cofactor \( C_{ij} \) is calculated as \( (-1)^{i+j} \) times the determinant of the submatrix formed by removing the ith row and jth column from \( A \). Since \( A \) is an upper triangular matrix, the cofactors are easy to calculate: \[ C = \begin{bmatrix} 24 & 0 & 0 \\ -30 & 6 & 0 \\ 20 & -5 & 1 \end{bmatrix} \]

Step 3 ：Finally, we find the transpose of the cofactor matrix to get the adjoint of \( A \): \[ \text{adj}(A) = C^T = \begin{bmatrix} 24 & -30 & 20 \\ 0 & 6 & -5 \\ 0 & 0 & 1 \end{bmatrix} \]