Gauss-Jordan Elimination Question

Question

Problem: Suppose that a matrix A satisfies

A

rref [-9 -43 * -133 *] [1 0 5 0 1]

[3 16 * 48 *] [0 1 -3 0 -2]

[5 25 * 76 *] = [0 0 0 1 1]

[-5 -24 * -74 *] [0 0 0 0 0]

Note that two of the columns in A are currently unknown.

Find the third column and fifth column of A without doing any row reductions.

Hi,

I'm really struggling to complete this problem and would greatly appreciate some assistance. I started with taking the column relations and setting them =0 but I'm not sure I'm on the right track. I believe I should be applying the Rouche Capelli Theorem but I'm not sure how. Please help me to understand how I should be solving this problem. Thank you.

Column relations:

col3 = 5col5 - 3col2 ---> col3-5col1+3col2 = 0

col5 = col1 - 2 col2 + col4 ---> col5 - col1 + 2col2 - col4 = 0

Answer 1

You are absolutely on the right track here with your column relations, since you know that column 5 of rref[A] is given by col1 - 2 col2 + col4, you can simply use the same column relations to determine that column 5 of A should also be given by col 1 - 2 col2 + col4 (however this time you use the columns of the matrix A). From there you should be able to find column 3 pretty easily.

You can do this because row operations preserve column relations. (It is a good idea to see if you can prove this yourself)

For further explanation/context, you do not need to use the Rouché-Capelli Theorem because you are not solving a system of linear equations; you are simply trying to determine what the original matrix A was from the information you have about its row reduced echelon form.