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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 $A$, the following statements are equivalent:
  1. $A$ is an invertible matrix.
  2. $A$ is row equivalent to the identity matrix $I_{n}$.
  3. $A$ is a product of elementary matrices.

Proof:

By Theorem 3, $A$ is row equivalent to a reduced row echelon matrix $R$, i.e. $R = E_{k} \cdots E_{1} \cdot A$. If $A$ is invertible, so is $R = E_{k} \cdots E_{1} \cdot A$. This implies that $R \cdot P = I_{n}$ for some invertible $P$. This further implies that each row of $R$ must be non-zero, hence $R$ must be $I_{n}$.

Next, if $R = I_{n} = E_{k} \cdots E_{1} \cdot A$, $\Rightarrow\;
E_{1}^{-1} \cdots E_{k}^{-1} \cdot I_{n} = A$. This shows that $A$ is a product of elementary matrices.

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

Q.E.D.

Corollary: If $A$ is an invertible matrix, then forming a new matrix $(A\mid I_{n})$ and performing row operations to $A$ 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 $A^{-1}$ of $A$.

Proof:

$
E_{k} \cdots E_{1} \cdot (A \mid I_{n}) =
(E_{k} \cdots E_{1} \cdot A \mid E_{k} \cdots E_{1} \cdot I_{n})
= (I_{n} \mid E_{k} \cdots E_{1})
$

Notice that $B \cdot A = I_{n}$, then $B \cdot A \cdot A^{-1} =
I_{n} \cdot A^{-1} = B \cdot I_{n}$.

Q.E.D.

Corollary: If $A, B$ are $m \times n$ matrices, then $A$ is row equivalent to $B$ if and only if $A = P \cdot B$ for some invertible matrix $P$.


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