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# Invertible Matrices

First, let us make an observation which will be used in the proofs of the next two theorems:

Observation: A square reduced row echelon matrix with no zero rows is an identity matrix.

Remark:

This is easily seen by considering at least'' and at most'' how many leading coefficients can occur on a square reduced row echelon matrix.

Theorem 8   For a square matrix , the following statements are equivalent:
1. is an invertible matrix.
2. is row equivalent to the identity matrix .
3. is a product of elementary matrices.

Proof:

By Theorem 3, is row equivalent to a reduced row echelon matrix , i.e. . If is invertible, so is . This implies that for some invertible . This further implies that each row of must be non-zero, hence must be .

Next, if , . This shows that is a product of elementary matrices.

Finally, since each elementary matrix is invertible, if is a product of elementary matrices, must be invertible.

Q.E.D.

Corollary: If is an invertible matrix, then forming a new matrix and performing row operations to to get its reduced row echelon form. If we obtain an identity matrix on the left hand side, then the right hand side is the inverse of .

Proof:

Notice that , then .

Q.E.D.

Corollary: If are matrices, then is row equivalent to if and only if for some invertible matrix .

Next: Application to the Solution Up: Systems of Linear Equations Previous: Square and Elementary Matrices   Contents   Index
Felix Hsu 2007-02-27