What Really IS a Matrix Determinant?
What Really IS a Matrix Determinant?

Image by Author (marcelmoos.com)

Thinking back to my high-school years, linear algebra was a topic I was particularly fascinated with. It gave me the skill to solve large systems of linear equations and a geometric perspective on the problem making the whole process intuitive.

However, regarding matrix determinants, I was taught that they are numbers for matrices, how to compute them, and not much more. It took until my university courses that I learned the beauty behind determinants.

As soon as I learned about determinants’ geometric meaning, I was wondering why this wasn’t already taught in high school as it is very easy to understand and mind-opening.

In mathematics, how to compute something should never be the first question. The first question is always: “What really IS it?”. Only then should we ask: “Ok now that we know what it is, how can we compute it”. Take derivates, for example, as most of us know what derivatives are:

Given a function, its derivative is its slope or rate of change.

It is such a simple description. Yet, defining derivatives in this way is so powerful and liberating. We understand what a derivative IS, regardless of the concrete function, or the dimensionality of the function, and regardless of how to compute it. The actual computation of derivatives differs a lot between different functions. However, derivatives’ fundamental meaning ties everything together and brings order into chaos.

No teacher would introduce derivatives to students like: “Given a function, a derivative is just another function and this is how you compute it…”. Still, for matrix determinants, such explanations seem to be widely spread. Defining determinants by their geometric meaning instead of just some numbers is just as powerful as thinking of derivatives as slopes instead of just as functions.

Determinants & Geometry

Before diving into determinants, let’s quickly recap what they are defined for: Matrices.

A matrix is a table of numbers, representing a linear function taking a vector as input and producing another vector as output:

Instead of a matrix transforming one single vector we can also think of a matrix transforming multiple (or even all) vectors at the same time:

Can you see it? It looks like our chosen matrix stretches space apart. Whatever area in the input space we choose, it seems that after the transformation the area gets bigger. This is precisely what the determinant is!

T he determinant of a matrix is the factor by which areas are scaled by this matrix.

Because matrices are linear transformations it is enough to know the scaling factor for one single area to know the scaling factor for all areas. Let’s go back to our example:

The rectangle inscribed by the pink and blue unit vectors and has an area of 1. After applying our matrix transformation, this rectangle has turned into a parallelogram with base 2 and height 2. So it has an area of 4. This means, that our matrix scales areas by a factor of 4. Therefore, the determinant of our matrix is 4. Neat, isn’t it?

There is one caveat to the story: Determinants can be negative! If we start with an area of 1 and scale it by a negative factor, we would end up with a negative area. And negative areas are nonsense. So how can we make sense of our nice geometric definition in the presence of negative determinants? Luckily the fix is straightforward: If a matrix has a negative determinant, let’s say -2, areas are scaled by 2. The minus just means that space reversed its orientation. “What does that now even mean?”, you might rightfully ask. Let’s take a look:

We can see that the given matrix scales areas by a factor of 2. If we look closely we further notice that the blue vector was on the right of the pink vector but ended up on the left side. This is what’s meant by “space reversed its orientation”. That’s why the determinant of the matrix is not 2 but -2. Including negative determinants we get the full picture:

T he determinant of a matrix is the signed factor by which areas are scaled by this matrix. If the sign is negative the matrix reverses orientation.

All our examples were two-dimensional. It’s hard to draw higher-dimensional graphs. The geometric definition of determinants applies for higher dimensions just as it does for two. In three-dimensional space, the determinant is the signed scaling factor for volumes and in even higher dimensions for hypervolumes.

With Knowledge Comes Power

Equipped with this new geometric definition of determinants we can solve things with ease which would be much harder to handle without it. For example, you might or might not have heard the following fact:

If a matrix has a determinant of 0 it is non-invertible.

A matrix being non-invertible means that the transformation the matrix represents cannot be undone or reverted. If we only know how determinants are computed and nothing about their geometric meaning, justifying this fact is tough. In contrast, using our freshly established intuition about determinants explaining why this is true becomes not that hard:

Let’s say we have a matrix with determinant 0. This means that the matrix scales all areas by a factor of 0, which in turn means that all areas become 0 after the transformation. This can only happen if the matrix squishes the whole space into a lower dimension. For example, the two-dimensional space would be squished into a single line or point and such a transformation cannot be undone.

Arriving at this point, we can be proud of ourselves. We introduced matrix determinants as area scaling factors and managed to justify a famous property of matrices and determinants. And we did all of this without even considering how determinants are computed. But that question should anyway be secondary.
['Marcel Moosbrugger']
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