Summary

For an $n\times n$ matrix A, the following are equivalent.

1.

A has a PLU-factorization.

2.

For any n-vector b, the system Ax = b has a solution.

3.

A has an inverse.

4.

A has non-zero determinant.

5.

A has full rank.

Read CLR Ch 31.1 for definitions of inverse, determinant and rank.