Properties of Elementary Matrices

For 2.

Invertible means that A is square and A^-1 exists or A is nonsingular.
A is nonsingular if it's determinant is nonzero.
So you need only show that every elementary matrix is square and has a nonzero determinant.

Use the fact that det(I) = 1. and that performing row operations on a matrix has the following results:
a. multiplying a row by a constant => multiply the determinant by that constant
b. interchanging rows => change sign of determinant
c. add multiple of one row to another row => no effect on determinant

Since every elementary matrix is the result of row operations on I, the determinant will be some multiple of 1 and thus nonzero.

To show that the inverse of an elementary matrix is elementary. Try showing that you can get to I by performing one row operation on E^-1.

For 1.
Here is a website that could be helpful. It outlines some of these properties and does several proofs that could be helpful.
http://tutorial.math.lamar.edu/Classes/LinAlg/InverseMatrices.aspx

You could use each type of row operation to generate an E. Then use that E on a generic A to show that EA has the same change as the one used to generate E.

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