Tuesday, October 9, 2018

Advanced Math Solutions - Matrix Row Echelon Calculator, Gaussian Elimination (Row Echelon Form)

In our previous blog posts, we talked about the Matrix basics. Now, we are ready to talk about a more advanced matrix topic, Gaussian Elimination (also known as row echelon form).

The Gaussian Elimination, is a method of putting a matrix in row echelon form (REF), using elementary row operations.

REF is when a matrix qualifies for the following two characteristics:
  • Each nonzero row has a leading coefficient (the first nonzero entry) that is to the right of the leading coefficient of the row above it
    • There can’t be any nonzero entries below the leading coefficient in the leading coefficient’s column
  • All zero rows are at the bottom

Note: A matrix in REF is not unique, so you may have a slightly different solution.

Here are examples of REF:




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


The elementary row operations you’ll use to put your matrix in REF are:
  • Switch any two rows
  • Multiply each entry in a row by a non-zero constant
  • Replace a row by the sum/difference of the row itself and another row, where it’s entries are multiplied by a non-zero constant

Here is a guideline on how to put a matrix in REF:
  1. Move all zero rows to the bottom
  2. Begin at the first row
  3. If the first entry is a zero, switch the row with a row below it that has non-zero entry in the first column
  4. If there are nonzero entries below the leading coefficient of the first row in the same column, cancel the entries by subtracting multiples of the the first row to the other rows (this will result in a zero entry)
  5. Repeat steps 2 - 4 and replace “first” with “second”, then “third”, and so on until you can’t do anything more

This topic can be hard to understand at first, so let’s see some examples to better understand.

First example (click here):


1.  Cancel the leading coefficient in Row 2: R_2-\frac{1}{3} R_1→ R_2


2. Simplify Row 2


We got rid of (canceled out) the 2 in Row 2 because there can’t be any entries below the leading coefficient in Row 1. We were able to do this using the elementary row operations. Multiplying Row 1 by ⅓, in order to turn 6 into 2, and then subtracting Row 2 by ⅓ Row 1.

Next example (click here):


1. Cancel the leading coefficient in Row 2: R_2-\frac{1}{2} R_1  →R_2


2. Cancel the leading coefficient in Row 3: R_3-(-\frac{3}{4} R_2) →R_3


Last example (click here):


1. Cancel the leading coefficient in Row 2: R_2-\frac{1}{2} R_1  →R_2


2. Cancel the leading coefficient in Row 3: R_3-\frac{1}{2} R_1→R_3


3. Since the second entry is a zero, switch the row with a row below it that has non-zero entry in the second column: R_3↔R_2



Putting matrices in REF can be tricky at first, but once you’ve done a handful of practice problems, it will come to you much easier. Check out Symbolab’s Row Echelon Calculator to help you better understand this topic.

Until next time,

Leah

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