# Homework Help: Reducing a matrix to echelon form

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1. Nov 11, 2017

### Mohamed Abdul

1. The problem statement, all variables and given/known data

(i) Reduce the system to echelon form C|d
(ii) For k = -12, what are the ranks of C and C|d? Find the solution in vector form if the system is consistent.
(iii) Repeat part (b) above for k = −18

2. Relevant equations

Gaussian elimination I used here:
http://mathworld.wolfram.com/GaussianElimination.html

3. The attempt at a solution

I've attempted reducing the matrix but have run into this problem where I have two zeroes leading in the last three rows. I don't know what I am doing wrong, so any help would be greatly appreciated.

I can't continue with either parts b or c without solving part a, and I'm not too sure about those parts either.

2. Nov 11, 2017

### NFuller

Your algebra looks ok. Having these zeros does not necessarily mean its wrong, and it is probably the reason that the problem is asking for the rank of the matrix in part ii.

3. Nov 11, 2017

### Mohamed Abdul

Alright, I understand. However, how would the ranks change regardless of my k value considering that the ranks only depend on the C part of C|d. Also, when it says to find the solution in vector form, does that mean I just write down the last column when I plug in k?

4. Nov 12, 2017

### NFuller

The rank of $C$ will not change. I am not completely sure what the question means when it asks for the rank of $C|d$.
You can write the solution vector as the last column $d$ if you get the augmented matrix into reduced echelon form. Otherwise, you can perform Gaussian elimination to find the solution.