The problem asks to multiply the matrices $A = \begin{bmatrix} 3 & 4 & 5 \end{bmatrix}$ and $B = \begin{bmatrix} 5 \\ 6 \\ 7 \end{bmatrix}$.

1. Problem Description

The problem asks to multiply the matrices $A = \begin{bmatrix} 3 & 4 & 5 \end{bmatrix}$ and $B = \begin{bmatrix} 5 \\ 6 \\ 7 \end{bmatrix}$.

2. Solution Steps

The matrix $A$ is a $1 \times 3$ matrix and the matrix $B$ is a $3 \times 1$ matrix. The product of two matrices $A$ and $B$ is defined only if the number of columns of $A$ is equal to the number of rows of $B$. In this case, the number of columns of $A$ is 3, and the number of rows of $B$ is 3, so the product $AB$ is defined. The resulting matrix will have the same number of rows as $A$ and the same number of columns as $B$, so the product $AB$ will be a $1 \times 1$ matrix.

To find the product $AB$, we multiply the entries of the first row of $A$ by the corresponding entries of the first column of $B$, and then we add the results.

$$$$

AB = \begin{bmatrix} 3 & 4 & 5 \end{bmatrix} \begin{bmatrix} 5 \\ 6 \\ 7 \end{bmatrix} = \begin{bmatrix} (3)(5) + (4)(6) + (5)(7) \end{bmatrix}

$$$$

$$$$

AB = \begin{bmatrix} 15 + 24 + 35 \end{bmatrix} = \begin{bmatrix} 74 \end{bmatrix}

$$$$

3. Final Answer

$\begin{bmatrix} 74 \end{bmatrix}$