Subsection PMI Properties of Matrix Inverses

The inverse of a matrix enjoys some nice properties. We collect a few here. First, a matrix can have but one inverse.

Theorem MIU Matrix Inverse is Unique

Suppose the square matrix $A$ has an inverse. Then $\inverse{A}$ is unique.

When most of us dress in the morning, we put on our socks first, followed by our shoes. In the evening we must then first remove our shoes, followed by our socks. Try to connect the conclusion of the following theorem with this everyday example.

Theorem SS Socks and Shoes

Suppose $A$ and $B$ are invertible matrices of size $n$. Then $AB$ is an invertible matrix and $\inverse{(AB)}=\inverse{B}\inverse{A}$.

Proof

Theorem MIMI Matrix Inverse of a Matrix Inverse

Suppose $A$ is an invertible matrix. Then $\inverse{A}$ is invertible and $\inverse{(\inverse{A})}=A$.

Proof

Theorem MIT Matrix Inverse of a Transpose

Suppose $A$ is an invertible matrix. Then $\transpose{A}$ is invertible and $\inverse{(\transpose{A})}=\transpose{(\inverse{A})}$.

Proof

Theorem MISM Matrix Inverse of a Scalar Multiple

Suppose $A$ is an invertible matrix and $\alpha$ is a nonzero scalar. Then $\inverse{\left(\alpha A\right)}=\frac{1}{\alpha}\inverse{A}$ and $\alpha A$ is invertible.

Proof

Notice that there are some likely theorems that are missing here. For example, it would be tempting to think that $\inverse{(A+B)}=\inverse{A}+\inverse{B}$, but this is false. Can you find a counterexample? (See Exercise MISLE.T10.)