Determinant

In this section, we will introduce the concept of the determinant of a matrix. It is a scalar value that can be thought of as a geometric scaling factor. It will be helpful to first understand the determinant in the context of 2D space before we generalize it to higher dimensions. Then, we will derive a formula to find this determinant.

Table of Contents

• Table of Contents
• Introduction
  • Negative Determinants
• Determinants in 3D
  • Orientation of 3D Space
• Deriving the 2D Determinant

Introduction

Imagine a 2D linear transformation that scales the -axis by a factor of and the -axis by a factor of . This transformation can be represented by the matrix:

Notice that this matrix appears to scale the space up:

Next, consider a different linear transformation:

This matrix appears to scale the space down:

Is there a way to quantify how much a matrix scales space? We can imagine placing a square in the space and observing how the matrix transforms it. For example, the first transformation scales the square by a factor of :

This factor, , is known as the determinant of the matrix .

However, we only know how the matrix scales this one unit square. What if we want to know how the matrix scales any shape in space?

• Since linear transformations preserve gridlines to be parallel and evenly spaced, any rectangle will get scaled by the same factor.
• Any shape can be approximated by a collection of rectangles.

Therefore, the determinant of a matrix represents how any shape in space is scaled by the matrix.

Below is a few examples of how different matrices scale space:

Different Determinants

Scale Up

Determinant:

This transformation scales space up by a factor of .

Scale Down

Determinant:

This transformation scales space down by a factor of .

Flip

Determinant:

This transformation does not scale space, but appears to "flip" it.

Shear

Determinant:

This shearing transformation, while it appears to distort space, does not scale it.

Rotate

Determinant:

Similar to the shearing transformation, this rotation does not scale space. In fact, all rotations have a determinant of .

Squish

Determinant:

Notice that this transformation squishes space to a line. This means that the square "collapses" to a line, which has no area, hence the determinant is .

Negative Determinants

In the example above, we saw that the determinant of a matrix can be negative if it "flips" space.

We can tell whether a space is flipped by looking at the basis vectors. Notice that in the normal case, the basis vectors are oriented in a counterclockwise direction such that is to the right of :

If this orientation is reversed, such that is to the left of , then the space is flipped. Consider the following transformation:

Notice that now, is to the left of ; this corresponds to a negative determinant.

Determinants in 3D

Consider extending the concept of determinants to 3D space.

In 2D, the determinant of a matrix represented how any area in space was scaled. We considered a unit square to understand this scaling. In 3D, we can instead consider a unit cube to understand how any volume in space is scaled.

Below is a 3D transformation along with a unit cube:

Orientation of 3D Space

In 2D, we could tell whether space was flipped by whether was to the left or right of . In 3D, we do something similar, but it is a bit more complicated, since we have to consider the orientation of three vectors.

The convention for this is known as the right-hand rule. Given three positively-oriented vectors , , and , if you curl your fingers from to , your thumb points in the direction of .

In 3D space, if you curl your fingers from to , your thumb should point in the direction of if space is positively oriented. If space is negatively oriented, your thumb will point in the opposite direction.

Positively Oriented

Negatively Oriented

Deriving the 2D Determinant

Now that we know what the determinant geometrically represents, we also need to be able to compute it.