Singular and Non-Singular Matrices

A square matrix $A$ is:

• singular if $|A|=0$;
• non-singular if $|A|\ne0$.

This one-number test carries a great deal of information about a square matrix — whether it can be 'undone' by another matrix, whether an associated system of linear equations has a unique solution, and so on (developed further in later courses on inverses).

Key fact. If $A$ and $B$ are non-singular matrices of the same order, then $AB$ and $BA$ are also non-singular, because $|AB|=|A|\,|B|=|BA|$ is a product of two nonzero numbers and hence itself nonzero.

Converse consequence. By the determinant product rule (§7.3.2, Property: $|AB|=|A||B|$), if $AB=O$ then $|A||B|=0$, so at least one of $A,B$ must be singular. A non-singular matrix can never combine with another matrix (via multiplication) to give the zero matrix — unless the other factor is already the zero matrix. …