The problem asks to multiply two matrices: a 1x3 matrix and a 3x1 matrix. We need to calculate the resulting 1x1 matrix. The given matrices are $\begin{bmatrix} 3 & 4 & 5 \end{bmatrix}$ and $\begin{bmatrix} 5 \\ 6 \\ 7 \end{bmatrix}$.

1. Problem Description

The problem asks to multiply two matrices: a 1x3 matrix and a 3x1 matrix. We need to calculate the resulting 1x1 matrix. The given matrices are $\begin{bmatrix} 3 & 4 & 5 \end{bmatrix}$ and $\begin{bmatrix} 5 \\ 6 \\ 7 \end{bmatrix}$.

2. Solution Steps

The multiplication of two matrices $A$ and $B$, where $A$ is an $m \times n$ matrix and $B$ is an $n \times p$ matrix, results in an $m \times p$ matrix. In this case, we have a $1 \times 3$ matrix multiplied by a $3 \times 1$ matrix, resulting in a $1 \times 1$ matrix (a scalar).

To find the element in the resulting matrix, we perform the dot product of the row of the first matrix and the column of the second matrix:

$C_{11} = (3)(5) + (4)(6) + (5)(7)$

$C_{11} = 15 + 24 + 35$

$C_{11} = 74$

Since the result is a 1x1 matrix, the only element is

7

4.

3. Final Answer

$74$