Gaussian Elimination - What am I doing wrong?

Question

I'm trying to solve a system of equations using Gaussian Elimiation but I'm really having some trouble as I seem to be getting too many 0's. Here are the equations:

+ + =

+ − =

+ + = 4

To get my first 0 in the bottom left corner, I have performed R3 - R2 + R1:

[1 1 1 = 1

4 3 -1 = 6

0 3 5 = -1]

Then to get the "0" on the second row, I perform R2 * R1 - R2:

[1 1 1 = 1

0  0  0 = 0

0 3 5 = -1]

Already I can see this is wrong as I have way too many 0's and can't see how I'll be able to get my diagonal 1's. My issue with this method is that I'm aware of what I'm allowed to do, but I'm not aware of what I'm not allowed to do.

Could someone please step me through a better way of performing Gaussian Elimitation with these equations?

Thank you in advance!

Answer 1

To obtain the RREF matrix we are allowed to do three types of elementary row operations:

1) Swapping two rows

2) Multiplying a row by a non-zero number

3) Adding a multiple of one row to another row

For example we can obtain

$$\begin{bmatrix}1&1&1&1\\4&3&-1&6\\3&5&3&4 \end{bmatrix} \stackrel{R3-3R1 + R2-4R1} \to\begin{bmatrix}1&1&1&1\\0&-1&-5&2\\0&2&0&1 \end{bmatrix} \stackrel{R3+2R2} \to\begin{bmatrix}1&1&1&1\\0&-1&-5&2\\0&0&-10&5 \end{bmatrix}$$

Answer 2

R2 * R1 - R2

This is not a valid elementary row operation. You can't multiply two rows together. You can only multiply them by (nonzero!) constants, or add and subtract them from each other, or swap them around.

Try something like $R_2 - 4R_1$ instead.