PROBLEM: Use as many columns of $A$ as possible to construct a matrix $B$ with the property that the equation $B x = 0$ has only the trivial solution.
Thoughts: Since I am doing this problem using Mathematica, I can pick and choose certain columns that will give me a matrix that reduces to Reduced Echelon Form. I chose columns $X_1$, $X_2$, $X_4$, and $X_5$. This new matrix $B$ row reduces completely, such that there is a pivot in each column.
Do I have the right idea? Is this what the question asked me to do?