How to choose the starting row when computing the reduced row echelon form?

I'm having hell of a time going around solving matrices to reduced row echelon form. My main issue is which row to start simplifying values and based on what? I have this example

image

so again, the questions are:

1.Which row to start simplifying values?

2.Based on what criteria?

Our professor solved it in the class with no fractions but I could not do it. Even though I know the 3 operations performed on matrices


Solution 1:

Where you start is not really a problem.

My tip:

  1. Always first make sure you make the first column: 1,0,0

  2. Then proceed making the second one: 0,1,0

  3. And lastly, 0,0,1

Step one:

$$\begin{pmatrix} 1&2&3&9 \\ 2&-1&1&8 \\ 3&0&-1&3\end{pmatrix}$$

row 3 - 3 times row 1

$$\begin{pmatrix} 1&2&3&9 \\ 2&-1&1&8 \\ 0&-6&-10&-24\end{pmatrix}$$

row 2 - 2 times row 1

$$\begin{pmatrix} 1&2&3&9 \\ 0&-5&-5&-10 \\ 0&-6&-10&-24\end{pmatrix}$$

Which simplifies to

$$\begin{pmatrix} 1&2&3&9 \\ 0&1&1&2 \\ 0&3&5&12\end{pmatrix}$$

Now you can proceed with step 2, and 3.

row 1 - 2 times row 2 and row 3 - 3 times row 2

$$\begin{pmatrix} 1&0&1&5 \\ 0&1&1&2 \\ 0&0&2&6\end{pmatrix}$$

Simplifies to

$$\begin{pmatrix} 1&0&1&5 \\ 0&1&1&2 \\ 0&0&1&3\end{pmatrix}$$

row 2 - row 3

$$\begin{pmatrix} 1&0&1&5 \\ 0&1&0&-1 \\ 0&0&1&3\end{pmatrix}$$

row 1 - row 3

$$\begin{pmatrix} 1&0&0&2 \\ 0&1&0&-1 \\ 0&0&1&3\end{pmatrix}$$

Solution 2:

The order of steps doesn't matter. Here are some tips:

1/ If the first row doesn't have 1 as the leading entry, make it be!

2/ Go by columns when you want to make entries becoming 0's. Usually, start from the first column and make all entries in the first column (except the leading 1 of the first row) be 0's.

3/ Avoid fractions. They create a big confusion when you apply elementary row/column operations. Sometimes you can multiply the fractional entries to become whole numbers, apply ERO's or ECO's, then reduce it back to be fraction. It's a bit redundant, but it helps you see what you have done.