1. **State the problem:** We are given a matrix $$A = \begin{bmatrix}4 & 0 & 0 \\ 0 & 6 & 0 \\ 8 & 0 & 5\end{bmatrix}$$ and we want to analyze it, for example, find its determinant or eigenvalues. 2. **Recall the determinant formula for a 3x3 matrix:** $$\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)$$ where the matrix is $$\begin{bmatrix}a & b & c \\ d & e & f \\ g & h & i\end{bmatrix}$$. 3. **Identify the elements:** $$a=4, b=0, c=0, d=0, e=6, f=0, g=8, h=0, i=5$$. 4. **Calculate the determinant:** $$\det(A) = 4(6 \times 5 - 0 \times 0) - 0(0 \times 5 - 0 \times 8) + 0(0 \times 0 - 6 \times 8)$$ $$= 4(30) - 0 + 0 = 120$$. 5. **Interpretation:** The determinant is 120, which is non-zero, so matrix $$A$$ is invertible. 6. **Optional: Eigenvalues** (since $$A$$ is upper triangular except for the 8 in the bottom left, eigenvalues are not simply diagonal elements, but for this matrix, we can find eigenvalues by solving $$\det(A - \lambda I) = 0$$). **Final answer:** The determinant of matrix $$A$$ is $$120$$.