Gauss-Jordan elimination is a terribly long and gruesome process but the results are very pretty so here’s my attempt to explain it:
Explanation applies to a 3x3 Matrix only! (you can adapt this process for other dimensions too)
Goal: To manipulate the matrix into the form above, having 1s in a diagonal running from the top left to the bottom right of the matrix and having 0 is every other spot.
This is because when you rewrite the matrix as an equation, the 0s ill eliminate other variables leaving the 1 with the variable that you’re trying to find. (this might seem confusing at first but try to follow through)
All about Manipulation! It’s almost like solving a rubiks cube having to work through each faces, layers, and corners.
1st Round Manipulation
Always start with the top left spot.
- We have to change the to right to be equal to the 1 0 0 format however
- You can only do change to the whole row and not a single spot.
- only multiplication/ division no subtraction/ addition
note: when referring to Row 1, it means the above
2. Row 2 and 3 Manipulation
why did I multiply row 1 by a factor of x5?
It prepares the first spot of Row 1 and the first spot of Row 2 for elimination.
If I add them together the first spot of Row 2 will equal zero which is getting closer to the ideal form that we want.
Notice that I do the same for Row 3, by multiplying Row 1 by a factor of 3 to set it up for elimination. Now when 3xRow 1 and Row 2 is subtracted the first spot of “the new Row 3” is now 0.
1st Round Rewrite
After the first column has been completely manipulated, do a rewrite to keep track of your updated matrix. (left hand, “where we are now”)
2nd Round Manipulation
question: what do I do next? Focus on the spot that has to be “1” in the second column and start manipulating that spot.
After dividing by 1.5, note that the “1” here becomes very important.
It is later on used to manipulate the other row’s middle term to make it 0.
2nd Round Rewrite
We’re 2/3 of the way there! The two columns are now equal so we’re left with the last column to work with.
3rd Round Manipulation
3rd Round Rewrite (Final Round)
Final Rewrite into Equation
the numbers in the matrix are the coefficient of the equation:
📌 Direct your frustrations at him: Carl Friedrich Gauss (just kidding or not?)
Yes it is a completely long and frustrating process so if you’re getting frustrated like I am / or was when I tried to learn this, direct your hate at this guy.