1. **Problem Statement:** Find the eigenvalues of the matrix expression $$3 + 8A - 5I$$ where $$A = \begin{bmatrix}4 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0\end{bmatrix}$$ and $$I$$ is the identity matrix. 2. **Recall:** The eigenvalues of a matrix expression involving $$A$$ and $$I$$ can be found by applying the expression to the eigenvalues of $$A$$. If $$\lambda$$ is an eigenvalue of $$A$$, then the corresponding eigenvalue of $$3 + 8A - 5I$$ is $$3 + 8\lambda - 5$$. 3. **Find eigenvalues of $$A$$:** Since $$A$$ is diagonal, its eigenvalues are the diagonal entries: $$\lambda_1 = 4, \quad \lambda_2 = 0, \quad \lambda_3 = 0$$. 4. **Calculate eigenvalues of $$3 + 8A - 5I$$:** - For $$\lambda_1 = 4$$: $$3 + 8(4) - 5 = 3 + 32 - 5 = 30$$ - For $$\lambda_2 = 0$$: $$3 + 8(0) - 5 = 3 - 5 = -2$$ - For $$\lambda_3 = 0$$: $$3 + 8(0) - 5 = -2$$ 5. **Final answer:** The eigenvalues of $$3 + 8A - 5I$$ are $$30, -2, -2$$.