# Table of Contents

# Definition

The determinant |**M**| of a matrix **M** is the product of the eigenvalues of **M**.

Given a volume in N-space with volume **V**, the image of the volume under **M** is
|**M**|V.

A matrix **M** is invertible if and only if |**M**| ≠ 0, and consequentially,
a matrix is invertible if and only if all the eigenvalues are nonzero.

# Determinants of Small Matrices

The determinant of the 2 × 2 matrix

may be found (referring to Figure 1) by simply multiplying the entries of the
diagonal (red arrow) and subtracting from this the product
of the entries on the anti-diagonal (blue arrow).

Figure 1. Determinant of a 2 × 2 matrix.

The determinant of the 3 × 3 matrix

may be found (referring to Figure 2) copying out the matrix
twice and taking the sum of the three products *aei* + *bfg* + *cdh*
(the three red arrows) and subtracting from this the sum
of the three products *ceg* + *afh* + *bdi* (the three blue arrows).

Figure 2. Determinant of a 3 × 3 matrix.

For example, the determinant of the matrix

is 3⋅5⋅4 + 1⋅(-1)⋅0 + (-2)⋅2⋅2 - (-2)⋅5;⋅0 - 3⋅(-1)⋅2 - 1⋅2⋅2

= 60 + 0 + (-8) - 0 - (-6) - 8 = 50.

**Note:** This does not generalize to higher matrices (4 × 4 and above).

## Example 1 (2 × 2)

The 2 × 2 matrix

has a determinant of |**M**_{1}| = 4 ⋅ 6 - 3 ⋅ 2 = 18. This is demonstrated
in Figure 3 which shows the image of the unit square (red) under multiplication
by **M**.

Figure 3. Image of the unit square under M_{1}.

## Example 2 (2 × 2)

The determinant of the 2 × 2 matrix

has a determinant of |**M**_{2}| = 3 ⋅ 4 - 6 ⋅ 2 = 0. Figure
4 shows how this maps a nonzero area (the unit square) onto a single line which has zero area. The
matrix is therefore non invertible, as there must be more than one point in the area which
is mapped to the same point on the line.

Figure 4. Image of the unit square under M_{2}.

## Example 3 (3 × 3)

The determinant of the 3 × 3 matrix

has a determinant of |**M**_{3}| = 50 (from above). Figure
5 shows how this maps the unit cube onto a 3-dimensional parallelepiped which has
area 50.

Figure 5. Image of the unit square under M_{3}.

## Example 4 (3 × 3)

The determinant of the 3 × 3 matrix

has a determinant of |**M**_{4}| = 1⋅5⋅9 + 2⋅6⋅7 + 3⋅4⋅8
- 3⋅5⋅6 - 1⋅6⋅8 - 2⋅4⋅9 = 0. Figure
6 shows how this maps a nonzero volume (the unit cube) onto a plane which has zero volume. The
matrix is therefore non invertible, as there must be more than one point in the volume which
is mapped to the same point on the plane.

Figure 6. Image of the unit square under M_{4}.

Copyright ©2005 by Douglas Wilhelm Harder. All rights reserved.