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# Section3.6The Invertible Matrix Theorem¶ permalink

##### 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 5.1.

##### 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.

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