Nonsingular matrices and their null spaces are intimately related, as the next two examples illustrate.
These two examples illustrate the next theorem, which is another equivalence.
Suppose that $A$ is a square matrix. Then $A$ is nonsingular if and only if the null space of $A$ is the set containing only the zero vector, i.e. $\nsp{A}=\set{\zerovector}$.
The next theorem pulls a lot of big ideas together. Theorem NMUS tells us that we can learn much about solutions to a system of linear equations with a square coefficient matrix by just examining a similar homogeneous system.
Suppose that $A$ is a square matrix. $A$ is a nonsingular matrix if and only if the system $\linearsystem{A}{\vect{b}}$ has a unique solution for every choice of the constant vector $\vect{b}$.
This theorem helps to explain part of our interest in nonsingular matrices. If a matrix is nonsingular, then no matter what vector of constants we pair it with, using the matrix as the coefficient matrix will always yield a linear system of equations with a solution, and the solution is unique. To determine if a matrix has this property (nonsingularity) it is enough to just solve one linear system, the homogeneous system with the matrix as coefficient matrix and the zero vector as the vector of constants (or any other vector of constants, see Exercise MM.T10).
Formulating the negation of the second part of this theorem is a good exercise. A singular matrix has the property that for some value of the vector $\vect{b}$, the system $\linearsystem{A}{\vect{b}}$ does not have a unique solution (which means that it has no solution or infinitely many solutions). We will be able to say more about this case later (see the discussion following Theorem PSPHS).
Square matrices that are nonsingular have a long list of interesting properties, which we will start to catalog in the following, recurring, theorem. Of course, singular matrices will then have all of the opposite properties. The following theorem is a list of equivalences.
We want to understand just what is involved with understanding and proving a theorem that says several conditions are equivalent. So have a look at Proof Technique ME before studying the first in this series of theorems.
Suppose that $A$ is a square matrix. The following are equivalent.
Finally, you may have wondered why we refer to a matrix as nonsingular when it creates systems of equations with single solutions (Theorem NMUS)! I have wondered the same thing. We will have an opportunity to address this when we get to Theorem SMZD. Can you wait that long?