Short Reduced Echelon Form Explanation?

[mneox]

Homework Statement

If my original echelon form is:

1 1 -2 1 | 2
0 3 3 3 | -3
0 0 0 1 | -4

and according to my notes that my teacher provided, the reduced form is:

1 0 -3 0 | 3
0 1 1 0 | 3
0 0 0 1 | -4

he noted that in the reduced form, the 1's in columns 1, 2, and 4 are pivots.

thanks for the help!

Or another question is, why is THIS in reduced row echelon form?

1 0 -2 0 | 9
0 1 -1 0 | 5
0 0 0 1 | 3

Homework Equations

none

The Attempt at a Solution

Now my question is why is the second 1 in row 2 not a pivot? why is there a -3 above the 1 in column 3? He stated that this matrix is in the reduced echelon form, so can someone just explain why its an rref if there's that -3?

Thanks for your help and time!

 

[Mark44]

Homework Statement

If my original echelon form is:

1 1 -2 1 | 2
0 3 3 3 | -3
0 0 0 1 | -4

and according to my notes that my teacher provided, the reduced form is:

1 0 -3 0 | 3
0 1 1 0 | 3
0 0 0 1 | -4

he noted that in the reduced form, the 1's in columns 1, 2, and 4 are pivots.

thanks for the help!

What is your question here?

Or another question is, why is THIS in reduced row echelon form?

1 0 -2 0 | 9
0 1 -1 0 | 5
0 0 0 1 | 3

How is the term "reduced, row-echelon form" defined in your book?

Homework Equations

none

The Attempt at a Solution

Now my question is why is the second 1 in row 2 not a pivot? why is there a -3 above the 1 in column 3? He stated that this matrix is in the reduced echelon form, so can someone just explain why its an rref if there's that -3?

In reduced, row-echelon form, a pivot is any leading entry of a row. The second 1 entry of row 2 is not a leading entry, so it isn't a pivot. It's only the leading entries (the pivots) for which the entries above and below are zero, so it doesn't matter that there is a nonzero entry above the second 1 entry in row 2.