(Pre-calc w/ Algebra) RREF for 3x3 Matrices

Question

I don't have much trouble putting a 2x2 matrix into reduced row echelon form, but I keep screwing up 3x3 matrices. EX from the current problem:

| 2 -17 11|

|-1 11 -7|

| 0 3 -2|

My issue is that I'll make the first numbers in rows 1, 2, and 3...1, 0, and 0 respectively but when I try to add and subtract rows, I lose my 1 in the first row and the issue continues when moving to the right.

My question: Is there a certain order that works best when reducing these numbers? maybe doing all zeros first? all ones first?

We've already moved on to inverse matrices which I'm understanding but can't do well in without knowing RREF haha.

Thanks in advance for the help!

Answer 1

Yes, there is an order.

Get a $1$ in the first column on the main diagonal if possible. Then use that $1$ to reduce all other entries in that column to $0$.

Then move to the next column and repeat, always getting the $1$ first and then using that $1$ to get $0$ in the other entries in that column.

Answer 2

You should perform row operations with a goal in mind.

The first goal could be to make the first column $(1,0,0)^T$, once you achieve that, once you achieve that, if you perform $R_1+cR_j$ wouldn't affect the the $1$ in the first row since $$R_{11}+cR_{j1}=1+c(0)=1$$

Also, you should not have operatins of the form of $R_j + cR_1$ because that would introduce a non-zero element in first element of row $j$.

Your goal should be to find the pivot elements.

For beginners: To perform Gauss-Jordan Elimination:

• Swap the rows so that all rows with all zero entries are on the bottom
• Swap the rows so that the row with the largest, leftmost nonzero entry is on top.
• Multiply the top row by a scalar so that top row's leading entry becomes $1$.
• Add/subtract multiples of the top row to the other rows so that all other entries in the column containing the top row's leading entry are all zero.
• Repeat steps $2$-$4$ for the next leftmost nonzero entry until all the leading entries are 1.
• Swap the rows so that the leading entry of each nonzero row is to the right of the leading entry of the row above it.

Reference can be found here.