Gauss-Jordan Elimination Method

Week 6: Section 2.3

We are given a system of linear equations (SLE). What is the first step in the Gauss-Jordan Elimination Method?

find the augmented matrix

Now that we have the augmented matrix, what is the second step in the Gauss-Jordan Elimination Method?

use the ERO to transform the augmented matrix into RREF

Once you have the matrix in RREF, what must you determine to find the solution?

if it has a unique, infinite, or no solutions

How do you know if the matrix has no solution?

If the matrix has a row with all zeros in the coefficient part, and a non-zero in the constant part

How can you tell if the matrix has a unique solution?

If every column of the coefficient part of the matrix has a leading 1

Once you determine that the matrix has a unique solution, how do you find the solution?

Turn the matrix into equation form and solve for the values of each variable

How can you tell when a matrix has infinitely many solutions?

When the matrix has 1 or more columns with no leading 1s

How do you solve for a matrix that has infinitely many solutions?

Assign parameters (t) to the columns without leading 1s and solve for the value of each variable in terms of (t)

In a matrix with infinitely many solutions, the number of parameters we assign is equal to what?

the number of columns without leading 1s

How do you turn a matrix into equation form

each column represents a variable, the value in each row represents the coefficient for that variable

Convert this matrix into an SLE

[ 1 0 3 ∣ 5 ]
[ 0 1 -2 ∣ 4 ]

x + 3z = 5

y -2z = 4

Does the following augmented matrix in RREF have a unique solution, no solutions or infinitely many solutions?

[ 1 0 0 | 5 ]
[ 0 1 0 | 3 ]
[ 0 0 1 | 1 ]

unique solution

Does the following augmented matrix in RREF have a unique solution, no solutions or infinitely many solutions?

[ 1 -3 0 -2 | 4 ]
[ 0 0 1 -1| 2 ]

infinitely many solutions

Does the following augmented matrix in RREF have a unique solution, no solutions or infinitely many solutions?

[ 1 0 0 | 9 ]
[ 0 1 0 | 6 ]
[ 0 0 0 | 1 ]

No solutions

The following augmented matrix in RREF has infinitely many solutions. How many parameters does it have?

[ 1 3 0 5 | 4 ]
[ 0 0 1 4 | 2 ]

2 parameters

Does the following augmented matrix in RREF have a unique solution, no solutions or infinitely many solutions?

[ 1 2 0 | 3 ]
[ 0 0 1 | 1 ]
[ 0 0 0 | 0 ]

infinitely many solutions