3 Matrices 3.4 Multiplication properties 3.6 Systems of linear equations

3.5 Invertible matrices

Definition 3.5.1.

An $n\times n$ matrix $A$ is called invertible if and only if there exists an $n\times n$ matrix $B$ such that $AB=BA=I_{n}$ .

If there is such a matrix $B$ , we can prove that there is only one such matrix $B$ :

Proposition 3.5.1.

If $AB=BA=I_{n}$ and $AC=CA=I_{n}$ then $B=C$ .

Proof.

$\displaystyle B$	$\displaystyle=BI_{n}$	Theorem 3.4.2
	$\displaystyle=B(AC)$
	$\displaystyle=(BA)C$	associativity
	$\displaystyle=I_{n}C$
	$\displaystyle=C$	Theorem 3.4.2

∎

This means that when a matrix is invertible we can talk about the inverse of $A$ . We write $A^{-1}$ for the inverse of $A$ when it exists.

3.5.1 Matrices with rows or columns of zeroes are not invertible

Theorem 3.5.2.

If an $n\times n$ matrix $A$ has a row of zeroes, or a column of zeroes, then it is not invertible.

Proof.

Suppose $A$ has a column of zeroes and that $B$ is any other $n\times n$ matrix. By Theorem 3.2.3, the columns of $B A$ are $B$ times the columns of $A$ . In particular, one of these columns is $B$ times the zero vector, which is the zero vector. Since one of the columns of $B A$ is all zeroes, $B A$ is not the identity.

If $A$ has a row of zeroes, we can make a similar argument using Theorem 3.2.6. ∎

3.5.2 Inverse of a product of matrices

If you multiply any number of invertible matrices together, the result is invertible. Recall the shoes-and-socks result about the inverse of a composition of two functions: exactly the same thing is true.

Theorem 3.5.3.

If $A_{1},\ldots,A_{k}$ are invertible $n\times n$ matrices then $A_{1}\cdots A_{k}$ is invertible with inverse $A_{k}^{-1}\cdots A_{1}^{-1}$ .

The proof is the same as for functions: you can simply check that $A_{k}^{-1}\cdots A_{1}^{-1}$ is a two sided inverse to $A_{1}\cdots A_{k}$ using the associativity property for matrix multiplication.

This theorem has a useful corollary about when matrix products are invertible.

Corollary 3.5.4.

Let $A$ and $E$ be $n\times n$ matrices with $E$ invertible. Then $E A$ is invertible if and only if $A$ is invertible, and $A E$ is invertible if and only if $A$ is invertible.

Proof.

If $A$ is invertible then the theorem tells us that so are $E A$ and $A E$ .

Suppose $E A$ is invertible. Certainly $E^{-1}$ is invertible (its inverse is $E$ ), so by the theorem $E^{-1}EA$ is invertible, that is, $A$ is invertible. The argument for $A E$ is similar. ∎

3.5.3 Inverse of a transpose

Theorem 3.5.5.

Let $A$ be an $n\times n$ matrix. Then $A$ is invertible if and only if $A^{T}$ is invertible.

Proof.

Suppose $A$ is invertible. We claim that $A^{T}$ is invertible with inverse $(A^{-1})^{T}$ . This is true because

$\displaystyle A^{T}(A^{-1})^{T}$	$\displaystyle=(A^{-1}A)^{T}$	Proposition 3.4.1 part 4
	$\displaystyle=I_{n}^{T}$
	$\displaystyle=I_{n}$

and similarly $(A^{-1})^{T}A^{T}=I_{n}$ .

Conversely, suppose $A^{T}$ is invertible. The above argument shows that $(A^{T})^{T}$ is invertible, that is, $A$ is invertible. ∎