Skill: Multiply a matrix by a column vector

To multiply a matrix by a column vector, check that the matrix has the same number of columns as the vector has entries. Then take the dot product of each matrix row with the vector. The results form a new column vector.

For example, calculate

$$\begin{pmatrix} 2 & -1 & 3\\ 0 & 4 & 1 \end{pmatrix} \begin{pmatrix} 5\\ 2\\ -1 \end{pmatrix}.$$

The matrix has $3$ columns, and the vector has $3$ entries, so the product is defined. Find each row’s dot product with the vector:

• First row: $2(5)+(-1)(2)+3(-1)=10-2-3=5$
• Second row: $0(5)+4(2)+1(-1)=0+8-1=7$

Therefore,

$$\begin{pmatrix} 2 & -1 & 3\\ 0 & 4 & 1 \end{pmatrix} \begin{pmatrix} 5\\ 2\\ -1 \end{pmatrix} = \begin{pmatrix} 5\\ 7 \end{pmatrix}.$$

The product has one entry for each row of the matrix.