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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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    Solution Preview

    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 ...

    Solution Summary

    This solution provides proof that for an elementary matrix, E, and arbitrary matrix,A; EA results in a matrix that is the same as if the row operation that created E was performed on A. Additionally, it provides proof that every E is invertible and the inverse of E is also an elementary matrix. The solution also includes a link to a source for further explanation of the concept.