Tuesday, October 23, 2018

Advanced Math Solutions - Matrix Gauss Jordan Reduction Calculator, Gauss Jordan Elimination

In our previous blog posts, we talked about Row Echelon Form (Gaussian Elimination). If you haven’t, please look over it before continuing with this blog post.

In this blog post, we’ll talk about another advanced matrix topic that uses the same concepts, Gauss Jordan Elimination.

The Gauss Jordan Elimination is a method of putting a matrix in row reduced echelon form (RREF), using elementary row operations, in order to solve systems of equations, calculate rank, calculate the inverse of matrix, and calculate the determinant of a matrix (we will cover this in the next few blog posts).

RREF is when a matrix qualifies for the following four characteristics:
  • Each non-zero row has 1, called a leading 1, as their first non-zero entry
  • Each column with a leading 1 has zeros in every other entry
  • As you move down the rows, the leading 1 moves to the right 
  • All zero rows are at the bottom

Another thing to note: Unlike matrices in REF, matrices in RREF are unique.

Here are examples of matrices in RREF:


Here are examples of matrices that aren’t in RREF:


You’ll use the same elementary row operations that you use to put a matrix in REF to put a matrix in RREF.

Here are guidelines on how to put a matrix in RREF:
  1. Put the matrix in REF
  2. If there are nonzero entries in the column of the leading coefficient in the first row, make them 0 by using the elementary row operations
  3. If the leading coefficient in the first row is not a 1, make it a 1 by multiply the row by the reciprocal (this turns the leading coefficient into the leading 1)
  4. Repeat steps 2-3 and replace “first” with “second”, then “third”, and so on

Gauss Jordan Elimination can be tricky the first few times, so I will walk you through 3 examples.

First example (click here):


1.  Put the matrix in REF


2.  Get rid of the nonzero entry in the column of the leading coefficient in Row 2: R_1-5R_2→R_1



3.  Turn the leading coefficient into a 1 in Row 1: \frac{1}{5} R_1→R_1


4.  Turn the leading coefficient into a 1 in Row 2: \frac{1}{2} R_2→R_2


Example 2 (click here):


1. Put the matrix in REF


2. Turn the leading coefficient in Row 1 into 1: \frac{1}{3} R_1→R_1


3. Turn the leading coefficient in Row 2 into 1: \frac{1}{8} R_2→R_2


In this problem there was no leading coefficient in Row 3, so we didn’t have to get rid of the entries in column 3.

Example 3 (click here):


1. Put the matrix in REF


2. Get rid of the nonzero entry in the column of the leading coefficient in Row 2: R_1-3R_2→R_1


3. Turn the leading coefficient in Row 1 into 1: \frac{1}{3} R_1→R_1


4. Turn the leading coefficient in Row 2 into 1: 3R_2→R_2


5. Turn the leading coefficient in Row 3 into 1: \frac{1}{2} R_3→R_3



Putting a matrix in RREF can require a lot of steps, so it is important to document what you are doing in each row. This will help you look over your work and remember what you did. Just like putting a matrix in REF, practice will help you get better at putting a matrix in RREF.

For more help or practice on this topic, visit Symbolab’s Practice.

Until next time,

Leah.

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