Subsection DNMMM Determinants, Nonsingular Matrices, Matrix Multiplication

If you asked someone with substantial experience working with matrices about the value of the determinant, they'd be likely to quote the following theorem as the first thing to come to mind.

Theorem SMZD Singular Matrices have Zero Determinants

Let $A$ be a square matrix. Then $A$ is singular if and only if $\detname{A}=0$.

Proof

For the case of $2\times 2$ matrices you might compare the application of Theorem SMZD with the combination of the results stated in Theorem DMST and Theorem TTMI.

Example ZNDAB Zero and nonzero determinant, Archetypes A and B

In Section MINM we said “singular matrices are a distinct minority.” If you built a random matrix and took its determinant, how likely would it be that you got zero?

Since Theorem SMZD is an equivalence (Proof Technique E) we can expand on our growing list of equivalences about nonsingular matrices. The addition of the condition $\detname{A}\neq 0$ is one of the best motivations for learning about determinants.

Theorem NME7 Nonsingular Matrix Equivalences, Round 7

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, $\nullity{A}=0$.
  11. The determinant of $A$ is nonzero, $\detname{A}\neq 0$.

Proof

Computationally, row-reducing a matrix is the most efficient way to determine if a matrix is nonsingular, though the effect of using division in a computer can lead to round-off errors that confuse small quantities with critical zero quantities. Conceptually, the determinant may seem the most efficient way to determine if a matrix is nonsingular. The definition of a determinant uses just addition, subtraction and multiplication, so division is never a problem. And the final test is easy: is the determinant zero or not? However, the number of operations involved in computing a determinant by the definition very quickly becomes so excessive as to be impractical.

Now for the coup de grace. We will generalize Theorem DEMMM to the case of any two square matrices. You may recall thinking that matrix multiplication was defined in a needlessly complicated manner. For sure, the definition of a determinant seems even stranger. (Though Theorem SMZD might be forcing you to reconsider.) Read the statement of the next theorem and contemplate how nicely matrix multiplication and determinants play with each other.

Theorem DRMM Determinant Respects Matrix Multiplication

Suppose that $A$ and $B$ are square matrices of the same size. Then $\detname{AB}=\detname{A}\detname{B}$.

Proof

It is amazing that matrix multiplication and the determinant interact this way. Might it also be true that $\detname{A+B}=\detname{A}+\detname{B}$? (Exercise PDM.M30)

Sage NME7 Nonsingular Matrix Equivalences, Round 7
Sage PDM Properties of Determinants of Matrices
(in context)