# Matrix Inverse

The inverse of a matrix should, like inverses of functions, operations, and numbers (arithmetic or multiplicative), satisfy . Here, a matrix inverse $M^{-1}$ is a *multiplicative* operation, so like the reciprocal for real numbers, it must satisfy $M \cdot M^{-1} = \text{Id}$ and $M^{-1} \cdot M = \text{Id}$. But what is this element, $\text{Id}$?

## The Identity

Remember that matrices were originally created to solve . It makes some sense, then, that the identity matrix with respect to matrix multiplication is the diagonal matrix with all $1$'s:

$\text{Id} = \begin{pmatrix} 1 & 0 & \dots & 0 \\ 0 & 1 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & 1 \end{pmatrix}.$

## Inverse Computations

The inverse of a matrix $M$ is calculated by row reducing a related matrix, namely the $n \times 2n$ matrix $[M \ I]$.

## Inverse Properties

The inverse of a matrix is used in many contexts throughout linear algebra, including similar matrices, diagonalizable matrices, and almost any discussion of linear transformations involving matrices.

It is therefore helpful to know a little bit more about the inverse of an invertible matrix $M$.

- The inverse of the inverse of a matrix is the matrix itself. That is, if $N = M^{-1}$, then $N^{-1} = M$.
- If $MN = \text{Id}$, then $M = N^{-1}$ and $N = M^{-1}$.
- The inverse commutes with transposition. That is, $(M^T)^{-1} = (M^{-1})^T$.
- Taking the inverse reverses the order of multiplication. That is, $(MN)^{-1} = N^{-1} M^{-1}.$