How to prove an elementary matrix is invertible?

Corkgirl18

I have a question that says Prove that an elementary matrix is invertible and that its inverse is again an elementary matrix?

Its probably a really easy question but the language is confusing me.
Any help is greatly appreciated.

Michael Collins

There are three types of elementary matrix. These are matrices which differ from the identity matrix by either

1) Rows swapped (interchange row i and j)
2) Row scaled (by a factor k not equal to zero)
3) Row addition (k times row j added to row i).

You should be able to find each of these in your notes if the above isn't immediately obvious to you.

Now, to prove what you are asked you will need to construct the inverse of each of the operations above. So for example, I will show an inverse exists for the first case above. If an elementary matrix E swaps rows i and j, then, to "undo" this operation we need to find the inverse - in this case this is easy, swapping again will clearly swap the rows back to the way they were - but because these are elementary matrices, that must be the identity matrix, so

EE = I

therefore the elementary matrix that swaps rows must have an inverse, and that inverse is itself and elementary matrix.

You should be able to give it a go now for the two remaining types 2) and 3).