# 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, \((MN)^{-1} = N^{-1} M^{-1}.\)