Answer
$\begin{bmatrix} 13\\25\end{bmatrix}$
Work Step by Step
Let us consider two matrices $A$ and $B$ and their multiplication can be possible when the number of columns of matrix $A$ is the same as the number of rows of the matrix $B$.
We are given that $A=\begin{bmatrix} 1 &2\\3&4\end{bmatrix} $ and $B=\begin{bmatrix} -1 \\7\end{bmatrix} $
We can see that the size of matrix $A$ is $2 \times 2$ and that of matrix $B$ is $2 \times 1$. This shows that the number of columns of matrix-$A$ is the same as the number of rows of the matrix-$B$. So, their product $AB$ can be computed.
Recall that if $A=\begin{bmatrix}a_{11} &a_{12}\\a_{21} &a_{22}\end{bmatrix}$ and $B=\begin{bmatrix} b_{11} \\b_{21}\end{bmatrix}$, then
\begin{align*}
AB=\begin{bmatrix}a_{11}b_{11} +a_{12}b_{21}\\
a_{21}b_{11} +a_{22}b_{21}\\
\end{bmatrix}
\end{align*}
Thus, using the formula above gives:
$AB=\begin{bmatrix} 1 &2\\3&4\end{bmatrix} \begin{bmatrix} -1 \\7\end{bmatrix}=\begin{bmatrix} 1(-1)+(2)(7)\\(3)(-1)+(4)(7) \end{bmatrix}=\begin{bmatrix} 13\\25\end{bmatrix}$
Therefore, $AB$ is equal to $\begin{bmatrix} 13\\25\end{bmatrix}$.
