Answer
The third column of $AB$ is the sum of the first two columns of $B$.
Work Step by Step
If $AB$ is defined for some matrix $A$, then by the definition of matrix multiplication, $AB=\begin{bmatrix}A\vec{b}_{1}&A\vec{b}_{2}&\cdots &A\vec{b}_{n}\end{bmatrix}$, where $B$ is assumed to have $n$ columns. Since $\vec{b}_{3}=\vec{b}_{1}+\vec{b}_{2}$, the distributivity of matrix multiplication over vector addition (Theorem 2(b); note that column vectors are simply $n\times 1$ matrices) allows us to conclude the following:
$\begin{align}AB&=\begin{bmatrix}A\vec{b}_{1}&A\vec{b}_{2}&A\vec{b}_{3}&\cdots &A\vec{b}_{n}\end{bmatrix}\\
&=\begin{bmatrix}A\vec{b}_{1}&A\vec{b}_{2}&A(\vec{b}_{1}+\vec{b}_{2})&\cdots &A\vec{b}_{n}\end{bmatrix}\\
&=\begin{bmatrix}A\vec{b}_{1}&A\vec{b}_{2}&A\vec{b}_{1}+A\vec{b}_{2}&\cdots &A\vec{b}_{n}\end{bmatrix}.
\end{align}$
This confirms that the third column of $AB$ is the sum of columns one and two.
