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

1. Problem Description

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

2. Solution Steps

To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. In this case, the first matrix is $1 \times 2$ and the second is $2 \times 2$, so the product will be a $1 \times 2$ matrix.

The first element of the resulting $1 \times 2$ matrix is obtained by multiplying the first row of the first matrix by the first column of the second matrix:

$(1 \times 5) + (2 \times 6) = 5 + 12 = 17$.

The second element of the resulting $1 \times 2$ matrix is obtained by multiplying the first row of the first matrix by the second column of the second matrix:

$(1 \times 7) + (2 \times 8) = 7 + 16 = 23$.

Therefore, the resulting matrix is $\begin{bmatrix} 17 & 23 \end{bmatrix}$.

3. Final Answer

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