Steps to Solve

1. Identify Matrix Dimensions Matrix $C$ is a $3 \times 3$ matrix, while Matrix $D$ is a $3 \times 2$ matrix. Matrix $A$ is also a $3 \times 3$ matrix.

2. Determine Compatibility for Multiplication To multiply matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Here, we can multiply $C$ ($3 \times 3$) with $D$ ($3 \times 2$) since the number of columns in $C$ (3) matches the number of rows in $D$ (2).

3. Matrix Multiplication of C and D The product of matrices $C$ and $D$ will form a new matrix $E$. The resulting matrix will have dimensions of $3 \times 2$.

Calculating each element of matrix $E$:

[ E_{ij} = \sum_{k=1}^{n} C_{ik} D_{kj} ]

For example, the element $E_{11}$ (first row, first column) is calculated as:

[ E_{11} = C_{11}D_{11} + C_{12}D_{21} = 1 \cdot 3 + 1 \cdot 2 = 5 ]

Continue calculating the remaining elements of $E$:

• For $E_{12}$: [ E_{12} = C_{11}D_{12} + C_{12}D_{22} = 1 \cdot -5 + 1 \cdot 1 = -4 ]

• For $E_{21}$: [ E_{21} = C_{21}D_{11} + C_{22}D_{21} = -1 \cdot 3 + 3 \cdot 2 = 3 ]

• For $E_{22}$: [ E_{22} = C_{21}D_{12} + C_{22}D_{22} = -1 \cdot -5 + 3 \cdot 1 = 8 ]

• For $E_{31}$: [ E_{31} = C_{31}D_{11} + C_{32}D_{21} = 2 \cdot 3 + 1 \cdot 2 = 8 ]

• For $E_{32}$: [ E_{32} = C_{31}D_{12} + C_{32}D_{22} = 2 \cdot -5 + 1 \cdot 1 = -9 ]

4. Construct Resulting Matrix E Now we can write matrix $E$:

[ E = \begin{bmatrix} 5 & -4 \ 3 & 8 \ 8 & -9 \end{bmatrix} ]

5. Matrix A and B Calculation (If Needed) If we need to perform operations with matrix $A$, ensure to check the dimensions first for compatibility for further operations.