For square matrices, the identity matrix plays the same role that the number $1$ plays in ordinary multiplication. It has ones on the main diagonal and zeros elsewhere:

$$I=\begin{pmatrix} 1&0&\cdots&0\ 0&1&\cdots&0\ \vdots&\vdots&\ddots&\vdots\ 0&0&\cdots&1 \end{pmatrix}.$$

Whenever the dimensions match,

$$AI=IA=A.$$

Inverse matrices

A square matrix $A$ is invertible when there is a matrix $A^{-1}$ such that

$$AA^{-1}=A^{-1}A=I.$$

The inverse undoes the action of $A$: applying $A$ and then $A^{-1}$ returns every vector to where it started.

Invertibility is not automatic

Not every square matrix has an inverse. For example, if a matrix sends two different input vectors to the same output, no matrix can uniquely undo that action.

If an inverse exists, it is unique. The notation $A^{-1}$ means the inverse matrix; it does not mean taking the reciprocal of each entry.

Products and inverses

If $A$ and $B$ are invertible, then their product is invertible and

$$(AB)^{-1}=B^{-1}A^{-1}.$$

The order reverses because the last action performed must be the first one undone.