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# Properties of Elementary Matrices

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Please prove the statements shown below:

1. If the elementary matrix E results from performing a certain row operation on an identity matrix Im and if A is an m x n matrix, then the product EA is the matrix that results when this same roe operation is performed on A.

2. Every elementary matrix is invertible, and the inverse is also an elementary matrix.

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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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