The problem asks us to find the product of two matrices, $C$ and $D$, given by: $C = \begin{bmatrix} 7 & 9 \\ 1 & 2 \end{bmatrix}$ $D = \begin{bmatrix} 2 & 8 \\ 3 & 7 \end{bmatrix}$ We need to compute the matrix $CD$.

1. Problem Description

The problem asks us to find the product of two matrices, $C$ and $D$, given by:

$C = \begin{bmatrix} 7 & 9 \\ 1 & 2 \end{bmatrix}$

$D = \begin{bmatrix} 2 & 8 \\ 3 & 7 \end{bmatrix}$

We need to compute the matrix $CD$.

2. Solution Steps

To find the product $CD$, we perform matrix multiplication. The element in the $i$-th row and $j$-th column of the product is obtained by taking the dot product of the $i$-th row of $C$ and the $j$-th column of $D$.

The resulting matrix will have the same number of rows as $C$ (which is 2) and the same number of columns as $D$ (which is 2).

Thus, $CD$ will be a $2 \times 2$ matrix.

$CD = \begin{bmatrix} 7 & 9 \\ 1 & 2 \end{bmatrix} \begin{bmatrix} 2 & 8 \\ 3 & 7 \end{bmatrix} = \begin{bmatrix} (7 \times 2 + 9 \times 3) & (7 \times 8 + 9 \times 7) \\ (1 \times 2 + 2 \times 3) & (1 \times 8 + 2 \times 7) \end{bmatrix}$

Now we calculate each element:

$7 \times 2 + 9 \times 3 = 14 + 27 = 41$

$7 \times 8 + 9 \times 7 = 56 + 63 = 119$

$1 \times 2 + 2 \times 3 = 2 + 6 = 8$

$1 \times 8 + 2 \times 7 = 8 + 14 = 22$

Therefore, $CD = \begin{bmatrix} 41 & 119 \\ 8 & 22 \end{bmatrix}$.

3. Final Answer

$CD = \begin{bmatrix} 41 & 119 \\ 8 & 22 \end{bmatrix}$