Inverses

Inverse of a Matrix
The "opposite" of a matrix or transformation, (inverse of ) has the property that applying , then gives the identity transformation. So, . Notes:

Computing the Inverse of a Matrix
Lemma 1. A square matrix is invertible if and only if it can be transformed by elementary row operations to row-reduced form with all diagonal entries nonzero.

Lemma 2. Any row-reduced square matrix with all diagonal elements nonzero can be transformed to the identity matrix with elementary row operations.

By these lemmas, we can write
so . So, we can compute the inverse of matrix by applying the elementary row operations to the identity matrix! So, to compute , apply the same row operations to the identity matrix as you applied to in order to transform it to the identity.
One way of finding the inverse of an matrix is as follows:

Ex.

It's good practice to verify results, so you should multiply the inverse by the original matrix to check that the result is the identity matrix.

As a side note, for matrices, we can find the determinant directly with the formula:

Invertible Matrix Theorem
For any matrix , the following are equivalent:

You don't need to memorize each one of these, but do take the time to understand why they are true — this will help you remember them.