1. The problem asks to find the eigenvalues of the matrix \( A = \begin{pmatrix} 3 & 2 & 1 \\ 0 & 4 & 2 \\ 0 & 0 & 1 \end{pmatrix} \) and then find the eigenvalues of the adjoint matrix \( \text{Adj } A \). 2. The adjoint \( \text{Adj } A \) of a matrix \( A \) is defined as the transpose of the cofactor matrix of \( A \). 3. Since \( A \) is an upper triangular matrix, its eigenvalues are the diagonal elements: $$ \lambda_1 = 3, \quad \lambda_2 = 4, \quad \lambda_3 = 1 $$ 4. The eigenvalues of \( \text{Adj } A \) are related to those of \( A \) by: $$ \text{If } \lambda_i \text{ are eigenvalues of } A, \text{ then the eigenvalues of } \text{Adj } A \text{ are } \prod_{j \neq i} \lambda_j $$ (This holds for invertible matrices; \( A \) has eigenvalues 3,4,1 so invertible.) 5. Calculate the eigenvalues of \( \text{Adj } A \): - For \( \lambda_1 = 3 \), adjoint eigenvalue is \( \lambda_2 \lambda_3 = 4 \times 1 = 4 \) - For \( \lambda_2 = 4 \), adjoint eigenvalue is \( \lambda_1 \lambda_3 = 3 \times 1 = 3 \) - For \( \lambda_3 = 1 \), adjoint eigenvalue is \( \lambda_1 \lambda_2 = 3 \times 4 = 12 \) 6. Therefore the eigenvalues of \( \text{Adj } A \) are: $$ 4, 3, 12 $$ 7. Summarizing: The matrix \( A \) has eigenvalues \( 3,4,1 \). The adjoint matrix \( \text{Adj } A \) has eigenvalues \( 4,3,12 \).