Skip to main content

Section4.6The Invertible Matrix Theorem

Objectives
  1. Theorem: the invertible matrix theorem.

This section consists of a single important theorem containing many equivalent conditions for a matrix to be invertible. This is one of the most important theorems in this textbook. We will append two more criteria in Section 6.1.

Proof

To reiterate, the invertible matrix theorem means:

There are two kinds of square matrices:

  1. invertible matrices, and
  2. non-invertible matrices.

For invertible matrices, all of the statements of the invertible matrix theorem are true.

For non-invertible matrices, all of the statements of the invertible matrix theorem are false.

The reader should be comfortable translating any of the statements in the invertible matrix theorem into a statement about the pivots of a matrix.

Now we can show that to check B = A 1 , it's enough to show AB = I n or BA = I n .

We conclude with some common situations in which the invertible matrix theorem is useful.