1.2 Row Reduction and Echelon Forms

Question 1

What is a non-zero row?

Answer 1

A row with at least one non-zero

Question 2

What is a leading entry?

Answer 2

The first non-zero number in a row

Question 3

Which three properties prove that a rectangular matrix is in row echelon form?

Answer 3

1. All non-zero rows are above all zero rows
2. Each leading entry of a row is in a column to the right of the leading entry in the row above it
3. All entries in a column below a leading entry are zeros

Question 4

what are the properties of a rectangular matrix in reduced echelon form?

Answer 4

The same three properties for row echelon form

4. Leading entry in each non-zero row is one
5. Each leading one is the only non-zero entry in its column

Question 5

What is echelon matrix or reduced echelon matrix?

Answer 5

A matrix in echelon form

Question 6

What does row reduced mean?

Answer 6

Any nonzero matrix may be reduced by elementary row operations into more than one matrix in echelon form

Question 7

What does an echelon form of A mean?

Answer 7

If matrix A is reduced to an echelon form say U

Question 8

What does the abbreviation RREF stand for?

Answer 8

Reduced row echelon form

Question 9

What are pivot positions?

Answer 9

Location in matrix with leading one in reduced echelon form

Question 10

What are pivot columns?

Answer 10

A column with a leading one

Question 11

What is a pivot?

Answer 11

A non-zero number in a pivot position that is used as needed to create zeros via row operations

Question 12

What does backward phase mean?

Answer 12

Reducing a matrix to reduce echelon form

Question 13

What does the forward phase mean?

Answer 13

Reducing a matrix to echelon form

Question 14

What does partial pivot mean?

Answer 14

When our computer program usually selects pivot with the largest absolute value

Question 15

What are basic variables?

Answer 15

The variable that corresponds to a pivot column in a coefficient matrix

Question 16

What are free variables?

Answer 16

Other variables that do not correspond to pivot columns

Question 17

What is a flop?

Answer 17

An arithmetic operation on two real floating point numbers

Question 17

What are parametric descriptions?

Answer 17

A form where free variables act as parameters

Question 18

What does theorem one say about the uniqueness of reduced row echelon form?

Answer 18

Each matrix is equivalent to one and only one reduced echelon matrix

Question 19

What is theorem two?

Answer 19

System is consistent if right most column of the matrix is a pivot column

Question 20

What are the steps of using row reduction to solve a linear system?

Answer 20

1. Write an augmented matrix of the system
2. Use a reduction algorithm to get an equal matrix in echelon form, decide whether the system is consistent, if there are no solutions stop
3. Continue row reduction to get reduced echelon form
4. write the system of equation to matrix achieved in step three
5. Rewrite each non-zero equation so basic variables are expressed in terms of any free variables