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Determinants

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 |M1| = 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 M1.

Example 2 (2 × 2)

The determinant of the 2 × 2 matrix

has a determinant of |M2| = 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 M2.

Example 3 (3 × 3)

The determinant of the 3 × 3 matrix

has a determinant of |M3| = 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 M3.

Example 4 (3 × 3)

The determinant of the 3 × 3 matrix

has a determinant of |M4| = 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 M4.

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