1. **Problem Statement:** We are given two 3x3 upper triangular matrices: $$A = \begin{bmatrix} a & 1 & 3 \\ 0 & e & 4 \\ 0 & 0 & 2 \end{bmatrix}, \quad B = \begin{bmatrix} \psi & r & 0 \\ 0 & 1 & \phi \\ 0 & 0 & f \end{bmatrix}$$ 2. **Goal:** Understand properties such as the product, determinant, or eigenvalues of these matrices. 3. **Key Properties of Upper Triangular Matrices:** - The determinant is the product of the diagonal elements. - The eigenvalues are the diagonal elements. - The product of two upper triangular matrices is also upper triangular. 4. **Determinants:** - For matrix $A$, $$\det(A) = a \times e \times 2 = 2ae$$ - For matrix $B$, $$\det(B) = \psi \times 1 \times f = \psi f$$ 5. **Eigenvalues:** - For $A$: $a$, $e$, and $2$ - For $B$: $\psi$, $1$, and $f$ 6. **Matrix Product $C = AB$:** Calculate $C = AB$ where $$C_{ij} = \sum_{k=1}^3 A_{ik} B_{kj}$$ Calculate each element: - $C_{11} = a\psi + 1 \times 0 + 3 \times 0 = a\psi$ - $C_{12} = a r + 1 \times 1 + 3 \times 0 = a r + 1$ - $C_{13} = a \times 0 + 1 \times \phi + 3 \times f = \phi + 3f$ - $C_{21} = 0 \times \psi + e \times 0 + 4 \times 0 = 0$ - $C_{22} = 0 \times r + e \times 1 + 4 \times 0 = e$ - $C_{23} = 0 \times 0 + e \times \phi + 4 \times f = e\phi + 4f$ - $C_{31} = 0 \times \psi + 0 \times 0 + 2 \times 0 = 0$ - $C_{32} = 0 \times r + 0 \times 1 + 2 \times 0 = 0$ - $C_{33} = 0 \times 0 + 0 \times \phi + 2 \times f = 2f$ So, $$C = \begin{bmatrix} a\psi & a r + 1 & \phi + 3f \\ 0 & e & e\phi + 4f \\ 0 & 0 & 2f \end{bmatrix}$$ 7. **Summary:** - Both $A$ and $B$ are upper triangular. - Their determinants are $2ae$ and $\psi f$ respectively. - Their eigenvalues are the diagonal elements. - The product $AB$ is also upper triangular with the matrix $C$ as above. This analysis helps understand matrix multiplication and properties of upper triangular matrices.