Mjollnir

Gaussian Elimination

Gaussian Elimination (or Gauss-Jordan Elimination) is usually the first procedure taught to students, partly because it can be presented to them as a logical outgrowth of simple substitution, and partly because the alternatives are much more complex. With Gaussian Elimination, the original matrix is augmented with an identity matrix, and the rows are added or subtracted to or from other rows, rows are multiplied by constants, and the rows are swapped. None of these operations affects the original problem, and students can imagine the same operations being applied to the original equations.

123 123
122-3100
2-102010
311-2001

Subtract row 3 from row 1 and add row 3 to row 2.

123 123
111-110-1
2010011
311-2001

Subtract row 1 from row 3.

123 123
111-11-1-1
2010011
300-1-102

Subtract row 2 from row 1.

123 123
110-11-1-1
2010011
300-1-102

Subtract row 3 from row 1.

123 123
11002-1-4
2010011
300-1-102

Multiply -1 through row 3.

123 123
11002-1-4
2010011
300110-2

The left side is now the identity matrix I and the right side is the inverse of the original A matrix (A-1). This procedure unnecessarily restricted the operations to simple additions and subtractions with a single sign-reversal, but might've been done faster by using products of rows and even swapping some rows.

To confirm that this is the inverse of the original matrix, we can multiply the inverse A-1 times the original matrix A and the original matrix A times the inverse A-1. In both cases the result should be the identity matrix (left- and right-hand inverse properties).

123 123 123
1 2-1-4 22-3 100
2 011 -102 = 010
3 10-2 11-2 001

and

123 123 123
1 22-3 2-1-4 100
2 -102 011 = 010
3 11-2 10-2 001

For the next part we'll do the same using nothing but column operations.