proof of matrix inverse calculation by Gaussian elimination

Let A be an invertible matrix, and A-1 its inversePlanetmathPlanetmath, whose columns are A1-1,,An-1. Then, by definition of matrix inverse, AA-1=In. But this implies AA1-1=e1,,AAn-1=en, with e1,,en being the first,,n-th column of In respectively.

A being non singular (or invertible), for all kn, AAk-1=ek has a solution for Ak-1, which can be found by Gaussian eliminationMathworldPlanetmath of [Aek].

The only part that changes between the augmented matrices constructed is the last column, and these last columns, once the Gaussian elimination has been performed, correspond to the columns of A-1. Because of this, the steps we need to take for the Gaussian elimination are the same for each augmented matrix.

Therefore, we can solve the matrix equation by performing Gaussian elimination on [Ae1en], or [AIn].

Title proof of matrix inverse calculation by Gaussian elimination
Canonical name ProofOfMatrixInverseCalculationByGaussianElimination
Date of creation 2013-03-22 14:15:10
Last modified on 2013-03-22 14:15:10
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 9
Author rspuzio (6075)
Entry type Definition
Classification msc 15A09