Matrice Echelons 2a: Check if Reduced & Find Single Op

[vorcil]

2a:
state whether each of the following matrices is in reduced echelon form. If it is not, then give a reason and say what single row operation is needed to bring it to reduced echelon form.


II)

1 1 0 3 0
0 0 1 2 0
0 0 1 2 1
not in reduced echelon form ( don't know how to do this one)
NOT SURE WHAT SINGLE OPERATION will bring to reduced echelon form

III)

1 0 -2 0
0 1 0 0
0 0 0 1
not in reduced echelon form(not sure how to remove the -2)
NOT SURE WHAT SINGLE OPERATION will bring to reduced echelon form

0 1 5 0 1
0 0 0 1 2
0 0 0 0 0
not in reduced echelon form (there is a 5 in the way)
NOT SURE WHAT SINGLE OPERATION will bring to reduced echelon form

VII)

1 4 0 1
0 0 0 0
0 0 1 3
not in reduced echelon form(there is a 4 in the way)
NOT SURE WHAT SINGLE OPERATION will bring to reduced echelon form

 

[Mark44]

2a:
state whether each of the following matrices is in reduced echelon form. If it is not, then give a reason and say what single row operation is needed to bring it to reduced echelon form.


II)

1 1 0 3 0
0 0 1 2 0
0 0 1 2 1
not in reduced echelon form ( don't know how to do this one)
NOT SURE WHAT SINGLE OPERATION will bring to reduced echelon form

Use the 2nd row to eliminate the 1 in the 3rd row.

III)

1 0 -2 0
0 1 0 0
0 0 0 1
not in reduced echelon form(not sure how to remove the -2)
NOT SURE WHAT SINGLE OPERATION will bring to reduced echelon form

This ***is*** in reduced row echelon form. Check your book to see how RREF is defined and I think you'll see why I'm saying you're done here.

0 1 5 0 1
0 0 0 1 2
0 0 0 0 0
not in reduced echelon form (there is a 5 in the way)
NOT SURE WHAT SINGLE OPERATION will bring to reduced echelon form

Again, check the definition of RREF. You're done here.

VII)

1 4 0 1
0 0 0 0
0 0 1 3
not in reduced echelon form(there is a 4 in the way)
NOT SURE WHAT SINGLE OPERATION will bring to reduced echelon form

Swap rows 2 and 3.

 

[Architeuthis Dux]

I would first define what reduced echelon form is in the context of matrices. Reduced echelon form is a specific form of a matrix where the leading coefficient (or pivot) of each row is 1, and all entries above and below each pivot are 0. Additionally, the pivots are arranged in a specific order, with each pivot to the right of the pivot in the row above it.

Based on this definition, I can determine that matrices II, III, and VII are not in reduced echelon form. In matrix II, there is a non-zero entry (1) above the pivot (1) in the third row, violating the rule that all entries above the pivot must be 0. To bring this matrix to reduced echelon form, a single row operation of multiplying the third row by -1 and adding it to the first row would eliminate the non-zero entry above the pivot.

In matrix III, there is a non-zero entry (-2) in the first row, violating the rule that the pivot must be 1. To bring this matrix to reduced echelon form, a single row operation of dividing the first row by -2 would make the pivot 1.

In matrix VII, there is a non-zero entry (4) above the pivot (1) in the second row, violating the rule that all entries above the pivot must be 0. To bring this matrix to reduced echelon form, a single row operation of dividing the second row by 4 would eliminate the non-zero entry above the pivot.

In summary, these matrices are not in reduced echelon form because they violate the rules of having a leading coefficient of 1 and all entries above and below the pivot being 0. To bring them to reduced echelon form, specific single row operations need to be performed to eliminate any non-zero entries above or below the pivot and make the pivot equal to 1.