We are asked to multiply two matrices. The first matrix is a $1 \times 2$ matrix $\begin{bmatrix} 1 & 2 \end{bmatrix}$ and the second matrix is a $2 \times 2$ matrix $\begin{bmatrix} 5 & 7 \\ 6 & 8 \end{bmatrix}$.

1. Problem Description

We are asked to multiply two matrices. The first matrix is a $1 \times 2$ matrix $\begin{bmatrix} 1 & 2 \end{bmatrix}$ and the second matrix is a $2 \times 2$ matrix $\begin{bmatrix} 5 & 7 \\ 6 & 8 \end{bmatrix}$.

2. Solution Steps

To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. In this case, the first matrix has 2 columns and the second matrix has 2 rows, so we can multiply them.

The result will be a $1 \times 2$ matrix.

Let the first matrix be $A = \begin{bmatrix} 1 & 2 \end{bmatrix}$ and the second matrix be $B = \begin{bmatrix} 5 & 7 \\ 6 & 8 \end{bmatrix}$.

The resulting matrix $C = A \times B$ will have elements:

$c_{11} = (1)(5) + (2)(6) = 5 + 12 = 17$

$c_{12} = (1)(7) + (2)(8) = 7 + 16 = 23$

Therefore, $C = \begin{bmatrix} 17 & 23 \end{bmatrix}$.

3. Final Answer

$\begin{bmatrix} 17 & 23 \end{bmatrix}$