Which are the pivot columns? - Linear Algebra

[tigrus]

Homework Statement

Given matrix

A=

1 1 -2 1 3
2 -1 2 2 6
3 2 -4 -3 -9

x=

x1
x2
x3
x4
x5

b =

1
2
3

1. Solve the system by elimination (use the augmented matrix) until pivots are found (no backward elimination here).

2. Show that pivots are respectively 1 and -3, and indicate clearly which are the pivot columns and free columns.

Homework Equations

Matrix A and vector b

The Attempt at a Solution

I perform elimination on augmented matrix to solve part 1.

1 1 -2 1 3 | 1
0 -3 6 0 0 | 0
0 -1 2 -6 -18 | 0


1 1 -2 1 3 | 1
0 -3 6 0 0 | 0
0 0 0 -6 -18 | 0

What I don't understand is part 2.

As far as I know, for an entry to become a pivot it must be the first non-zero entry on the row and there must be no non-zero entries under it.

1 1 -2 1 3 | 1
0 -3 6 0 0 | 0
0 0 0 -6 -18 | 0

If that rule applies I would find three pivots instead of two. (1,-3, -6) I know the right answer for this exercise is that there are only two pivots (1, -3), resulting col1 & col2 to be the pivot columns and the rest are free columns.

Can someone explain why -6 is not a pivot in this case?

 

[Mark44]

Homework Statement

Given matrix

A=

1 1 -2 1 3
2 -1 2 2 6
3 2 -4 -3 -9

x=

x1
x2
x3
x4
x5

b =

1
2
3

1. Solve the system by elimination (use the augmented matrix) until pivots are found (no backward elimination here).

2. Show that pivots are respectively 1 and -3, and indicate clearly which are the pivot columns and free columns.

Homework Equations

Matrix A and vector b

The Attempt at a Solution

I perform elimination on augmented matrix to solve part 1.

1 1 -2 1 3 | 1
0 -3 6 0 0 | 0
0 -1 2 -6 -18 | 0


1 1 -2 1 3 | 1
0 -3 6 0 0 | 0
0 0 0 -6 -18 | 0

What I don't understand is part 2.

As far as I know, for an entry to become a pivot it must be the first non-zero entry on the row and there must be no non-zero entries under it.

1 1 -2 1 3 | 1
0 -3 6 0 0 | 0
0 0 0 -6 -18 | 0

If that rule applies I would find three pivots instead of two. (1,-3, -6) I know the right answer for this exercise is that there are only two pivots (1, -3), resulting col1 & col2 to be the pivot columns and the rest are free columns.

Can someone explain why -6 is not a pivot in this case?

I don't see an error on your part. Are you sure you have copied the matrix correctly?

 

[CornMuffin]

as far as I could tell, there should be 3 pivots in your reduced matrix.

I suspect that you wrote the matrix down wrong, because if you make the last row
-3 3 -4 -3 -9 instead of 3 3 -4 -3 -9, then there should be only two pivots.