The problem asks us to multiply the given matrices. The first matrix is a $1 \times 3$ matrix, and the second matrix is a $3 \times 1$ matrix. We are asked to compute the product of these two matrices. The matrices are $\begin{bmatrix} 3 & 4 & 5 \end{bmatrix}$ and $\begin{bmatrix} 5 \\ 6 \\ 7 \end{bmatrix}$.

1. Problem Description

The problem asks us to multiply the given matrices. The first matrix is a $1 \times 3$ matrix, and the second matrix is a $3 \times 1$ matrix. We are asked to compute the product of these two matrices.

The matrices are $\begin{bmatrix} 3 & 4 & 5 \end{bmatrix}$ and $\begin{bmatrix} 5 \\ 6 \\ 7 \end{bmatrix}$.

2. Solution Steps

To multiply a $1 \times 3$ matrix by a $3 \times 1$ matrix, we perform the following calculation:

$\begin{bmatrix} a & b & c \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = [ax + by + cz]$

In our case, we have:

$\begin{bmatrix} 3 & 4 & 5 \end{bmatrix} \begin{bmatrix} 5 \\ 6 \\ 7 \end{bmatrix} = [ (3 \times 5) + (4 \times 6) + (5 \times 7) ]$

Calculating the products:

$3 \times 5 = 15$

$4 \times 6 = 24$

$5 \times 7 = 35$

Adding these results:

$15 + 24 + 35 = 74$

Therefore, the result of the matrix multiplication is a $1 \times 1$ matrix containing the value

7

4.

3. Final Answer

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