proof of matrix inverse calculation by Gaussian elimination
Let be an invertible matrix, and its inverse, whosecolumns are .Then, by definition of matrix inverse, .But this implies,with being the first,,-th column of respectively.
being non singular (or invertible), for all , has a solution for , which canbe found by Gaussian elimination of .
The only part that changes between the augmented matricesconstructed is the last column, and these last columns, once theGaussian elimination has been performed, correspond to the columnsof . Because of this, the steps we need to take for theGaussian elimination are the same for each augmented matrix.
Therefore, we can solve the matrix equation by performing Gaussianelimination on , or .