Gauss-Jordan Elimination Problem

Question

Hello,

I am struggling to understand Gauss-Jordan Elimination. This question I believe is asking for me to apply the Rouche-Capelli Theorem. I would greatly appreciate your guidance on how to approach this problem. Thank you.

Problem Statement: The matrix A is rank one. Find rref(A) without doing any row reductions. Hint: Start by identifying the column relations.

The matrix A =

[3 15 -24]

[-5 -25 40]

[2 10 -16]

[-7 -35 56]

Note: A is a (4 by 3) matrix

How I started to solve:

col2 = 5 × col1 --> col2 -5 × col1 = 0

col3 = -8 × col1 --> col3 + 8 × col1 = 0

[-5 8]

[1 0]

[0 1]

Not sure what to do next...

Answer 1

Since the rank is 1, all columns must be related.

Note that column 2 is 5 times column 1, and column 3 is -8 times column 1. So you could reduce columns 2 and 3 to all zeros and you would be done, without any complicated calculations.