Determinant: Intuition

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.

This page will focus on the geometric intuition behind the determinant. The next page will dive into the derivation for the formula to compute determinants.

Table of Contents

• Table of Contents
• Introduction
  • Negative Determinants
• Determinants in 3D
  • Orientation of 3D Space
• Implications on Linear Transformations
• Multiplied Determinants
• Summary and Next Steps

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

Implications on Linear Transformations

Consider a linear transformation that maps everything in 2D space to a single line on that space. We have already seen that this transformation can be represented by a matrix whose columns are linearly dependent.

Since the determinant of a matrix represents how space is scaled, then if the transformation squishes space to a line, the determinant must be ; the line has no area:

It also works the other way around: if the determinant of a matrix is , then the transformation must squish space to either a single point (the origin) or a line. In both cases, the columns of the matrix are linearly dependent.

Hence:

The determinant of a matrix is if and only if the columns of the matrix are linearly dependent.

Additionally, when the determinant is , it also means that there are multiple vectors that get mapped to the same vector. Take the same transformation as above, and consider the following vectors in the space:

We say that this transformation has a mapping that is not one-to-one (or injective) because multiple vectors get mapped to the same vector. This will be very important when we discuss the concept of invertibility of matrices.

Multiplied Determinants

When we multiply two matrices, the determinant of the product is the product of the determinants of the individual matrices.

We can formally prove this, but there's a very simple geometric intuition behind this.

Recall that multiplying two matrices is equivalent to applying two transformations in sequence, called a composite transformation. The determinant of a matrix represents how space is scaled by that matrix.

When we multiply two matrices, we are applying two transformations in sequence. If the first transformation scales space by , and the second transformation scales space by , then the composite transformation scales space by . Following this logic, the first transformation, , scales space by , and the second transformation, , scales space by . Then the overall transformation scales space by .

Summary and Next Steps

In this section, we introduced the concept of the determinant of a matrix.

Here are the key points to remember:

• The determinant of a matrix represents how space is scaled by the matrix.
• The determinant is positive if space is not "flipped", and negative if it is.
• A space can be seen as "flipped" if the orientation of the basis vectors is reversed. In 2D, this is when is to the left of . In 3D, this is determined by the right-hand rule.
• The determinant of a matrix is if and only if the columns of the matrix are linearly dependent.

In the next section, we will actually derive the formula to compute the determinant of a matrix.