Answer
Because there is a linear combination of A such that it is 0, the columns of A are linearly dependent
We know that the right-most column of A is all zero.
Work Step by Step
Here is a generalized depiction. Suppose \[
A=
\left[ {\begin{array}{cc}
l & m \\
n & p \\
\end{array} } \right]
\] and \[
B=
\left[ {\begin{array}{cc}
a & b \\
c & d \\
\end{array} } \right]
\]
Then
\[
AB=
\left[ {\begin{array}{cc}
la+mc & md+lb \\
na+pc & nb+pd \\
\end{array} } \right]
\]
Every value of A must be 0. For the second column of AB to be 0, we have a system of equations:
0=md+lb
0=nb+pd
0=(m+p)d+(l+n)b. Neither b nor d is 0.
We can declare that the sum of the values in each column are either all 0 or that the sums of the values for each column must be equal and opposite of the other.
