Subsection RNNM Rank and Nullity of a Nonsingular Matrix

Let us take a look at the rank and nullity of a square matrix.

Example RNSM Rank and nullity of a square matrix

The value of either the nullity or the rank are enough to characterize a nonsingular matrix.

Theorem RNNM Rank and Nullity of a Nonsingular Matrix

Suppose that $A$ is a square matrix of size $n$. The following are equivalent.

  1. A is nonsingular.
  2. The rank of $A$ is $n$, $\rank{A}=n$.
  3. The nullity of $A$ is zero, $\nullity{A}=0$.

Proof

With a new equivalence for a nonsingular matrix, we can update our list of equivalences (Theorem NME5) which now becomes a list requiring double digits to number.

Theorem NME6 Nonsingular Matrix Equivalences, Round 6

Suppose that $A$ is a square matrix of size $n$. The following are equivalent.

  1. $A$ is nonsingular.
  2. $A$ row-reduces to the identity matrix.
  3. The null space of $A$ contains only the zero vector, $\nsp{A}=\set{\zerovector}$.
  4. The linear system $\linearsystem{A}{\vect{b}}$ has a unique solution for every possible choice of $\vect{b}$.
  5. The columns of $A$ are a linearly independent set.
  6. $A$ is invertible.
  7. The column space of $A$ is $\complex{n}$, $\csp{A}=\complex{n}$.
  8. The columns of $A$ are a basis for $\complex{n}$.
  9. The rank of $A$ is $n$, $\rank{A}=n$.
  10. The nullity of $A$ is zero, $\nullity{A}=0$.

Proof
Sage NME6 Nonsingular Matrix Equivalences, Round 6
(in context)