Question

How to find inverse of a matrix using elementary transformation?

Answer 1

Answer:

To get the inverse of A, first make the augmented matrix

[ A | I ]

with A on the left and the identity matrix I on the right.

Then perform elementary row operations on this matrix until A is transformed into the identity:

[ I | B ]

The matrix B on the right is now the inverse of A.

To see why this works, notice that by performing elementary row operations, we're really just multiplying on the left by elementary matrices.  Say that all together, the row operations have amounted to multiplying by the matrix P.  Then we have

P [ A | I ] = [ I | B ]  =>  [ PA | P ] = [ I | B ]

In particular, PA = I, so P is the inverse of A, and P = B, so B is the inverse of A.