What are some real-world uses of the determinant of a matrix?

My answer to a question on Quora:

What are some real-world uses of the determinant of a matrix?

At the time of writing, I am engaged in a small debate with a colleague on one of the LinkedIn discussion groups: he teaches students to solve systems of 2 linear equations with 2 variables using Cramer’s rule (that is, via determinants), without giving any justification or proof for it, but I personally prefer self-justified solutions: for systems of 2 linear equations with 2 variables, the honest Gaussian elimination is quick, and it is easy to explain to students why it gives the right solution. Moreover, every intermediate step of Gaussian elimination can be naturally interpreted in terms of the original system of equations.

And this goes to the heart of the matter: in life, determinants are almost never used in computation. Someone said that

mathematics is the art of avoiding calculations;

in that sense,

linear algebra is the art of avoiding calculations with matrices,

and the rule of thumb is

avoid calculations with determinants!

For example, you can invert a matrix in essentially the same time as compute its determinant; after that the use of the cofactor formula for the inverse of a matrix and Cramer’s rule for solving systems of linear equations becomes waste of time.

However, determinants provide extremely efficient tools for thinking about problems of linear algebra, including those in practical applications. Linear algebra in its development or exposition goes through more and more compressed expression of relevant mathematical meaning, and the value of the determinant: zero or not zero is perhaps the most compressed form of expression of linear dependence / independence of \(n\) vectors in \(\mathbbR^n\).

Determinants have wonderful algebraic properties and occupied their proud place in linear algebra because of their role in higher level algebraic thinking.

In this thread on Quora some uses of determinants were mentioned, for example, computation of eigenvalues of a matrix; I am not an expert in numerical linear algebra, but I have a feeling that most methods for computation of eigenvalues do not even mention the word “determinant”. Even at a theoretical level, determinants can be excluded from the standard treatment of linear algebra, see Sheldon Axler’s paper Down with Determinants!

So, let me summarise:

• If you need more that just application of existing computer programs for solving practical problems of linear algebra and have to think about the process of solution, you may find determinants very useful indeed.
• Determinants can be meaningfully used for compact formulation of mathematical models of physical phenomena (perhaps this applies not only to physics). This thread in Quora contains some nice examples.
• But is is best avoid calculation of determinants.