How to multiply a matrix by a vector

We can write a vector as a column matrix, which lets us multiply it by another matrix using the same row-by-column process. Each row produces one component of the answer. Let’s work through a product and see what changes when we reverse the order.

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Check the dimensions first

An $m\times n$ matrix multiplies an $n$-component column vector to produce an $m$-component column vector. Use

$M=\left[\begin{array}{rrr}4& -2& 1\\ 0& 3& 0\\ -6& 1& -2\end{array}\right]$

$v=\left[\begin{array}{r}-2\\ 1\\ 3\end{array}\right]$

Use one row to calculate each component

$\begin{array}{l}(Mv{)}_1=4(-2)-2(1)+1(3)=-7\\ (Mv{)}_2=0(-2)+3(1)+0(3)=3\\ (Mv{)}_3=-6(-2)+1(1)-2(3)=7\end{array}$

$Mv=\left[\begin{array}{r}-7\\ 3\\ 7\end{array}\right]$

Use a row vector to multiply on the left

If $v$ is written as the row vector $[-2,1,3]$, we can multiply $vM$. This time the single row combines with each column of $M$, giving

$vM=\left[\begin{array}{rrr}-26& 10& -8\end{array}\right]$

The order and the way we write the vector determine which components are paired.

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