# Construct a matrix $B$ with the property that the equation $Bx = 0$ has only the trivial solution

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?

• Yes, you are correct. That is exactly what they want.
– jgon
Jan 22, 2018 at 5:48
• What are the sizes of the matrix B and x?If you form the matrix B with four columns of A such that B is invertible then the system Bx=0 will always have trivial solution. Jan 22, 2018 at 6:28
• Based on what you did you've already shown how to do it with 4 columns. What remains to be argued is why it cannot work with 5 columns. Though that is almost trivial. Jan 22, 2018 at 13:54

What you did is correct. You are searching for a matrix with kernel/null space equal to $\{0\}$. Here is the idea behind your answer.
Now you found a matrix that has reduced echelon form $I$, which means it is invertible, thus bijective, so only $x=0$ is the solution to $Bx=0$.