A Geometric Demonstration of the Formula of the Determinant of a Three-by-Three Matrix

By Zimo Chen, Year 12 Student

1. Introduction

Matrices are a powerful mathematical tool that are widely used in the STEM subjects. They are effective in describing linear transformations. For example, in optical physics, the Ray Transfer Matrix Analysis (RTMA), a mathematical approach for ray tracing, provides simple descriptions in the form of a Ray Transfer Matrix (RTM) on the change of direction of the light beam when undergoing different optical effects, including propagation, reflection, refraction and other optical behaviours (some are listed in Table 1 below), where light is assumed to be a ray when it moves through the same uniform medium[1]. Matrices are also used to describe and control spatial movements through the use of vectors and programmed transformations[2].

The determinant is an important value of a square matrix, commonly denoted det(M) or |M|. In the case of RTMA, it is a property that the determinant of the resultant transformation matrix — obtained by multiplying all the transformation matrices—corresponds to the overall index of refraction[1]. In linear algebra, the determinant is most often used to calculate the inverse matrix and solve equations with Cramer’s rule, where the geometric isomorphism of matrices is the underlying principle. The determinant also determines whether a given matrix is inversible (when det(M) ≠ 0) or not (when det(M) = 0).

This article aims to demonstrate a geometric deduction of the formula of the determinant of a three-by-three matrix by finding the volume of a unit parallelepiped from the vertex coordinates.

2. Algebraic Method of Calculating the Determinant

The determinant of a square matrix is a scalar value that can be calculated from the entries of a matrix. For example, for an arbitrary one-by-one matrix, M1, its determinant is its sole entry,

For an arbitrary two-by-two matrix, M2, its determinant can be calculated by

Similarly, the formula of the determinant of a three-by-three matrix, M3, is

The determinant of an n by n matrix can also be calculated using the Laplace expansion, where the determinant of the n by n matrix is a combination of the determinants of its submatrices along any one of its rows or columns, where in the case of a three-by-three matrix, it is

3. Geometric Interpretations

3.1 Introduction to the Geometric Method: Finding Areas

Along with the algebraic definitions of the determinant of two-by-two and three-by-three matrices mentioned above, the determinant also incorporates a geometric interpretation related to the linear transformation described by the square matrix. For a two-by-two matrix,

where A is the signed area (i.e. area with a + or − signed) of an arbitrary closed shape (i.e. any arbitrary shape whose ending and beginning are at the same exact point[4]), and A′ is the signed area of that shape after the transformation.

The sign of det(M) gives a clue to the orientation of the transformed coordinate system. Since the position vector of any point in a coordinate system — both before and after a linear transformation — can be expressed as a unique combination of the unit vectors, we can calculate the ratio of areas by focusing on the unit square. Specifically, we consider the parallelogram formed by the two unit vectors (along the x- and y-axes) as adjacent sides, both before and after the transformation. After the transformation, the new positions of the unit vectors (relative to the original coordinate system) are given by the columns of the transformation matrix. For example, let

be an arbitrary two-by-two matrix (the entries of the matrix are assumed to be real as geometry with complex numbers is not consider in this article.). The position vectors of the unit vectors after the transformation referred to the original coordinate system would be

where î' is the unit vector along the positive x direction and ĵ' is the unit vector along the positive y direction.

The signed area of the parallelogram can be calculated using simple geometry (Figure 1), which is equivalent to the determinant of M2.

Figure 1: Calculating the area of the parallelogram defined by the unit vectors  and , which is equal to the determinant of the two-by-two matrix represented by the transformation. This method bypasses the use of the Pythagorean theorem and avoid computing square roots and dot products.

The general idea is the same for finding the formula of the determinant of a three-by-three matrix geometrically, but instead of a parallelogram, the determinant is represented by the signed volume ratio of a 3D shape before and after the transformation described by the matrix[5]. Similar to how the position vector of every point in the x — y coordinate system can be represented by a unique combination of î' and ĵ, every position vector of the points in the x — y — z space is a unique combination of î, ĵ, and k̂ (k̂ is the unit vector along the positive z direcion Therefore, the signed volume of the “unit parallelepiped”, where the three unit vectors are three non-parallel edges, after the transformation is a geometric isomorphism of det(M3).

3.2 Calculating Volumes

The volume of the parallelepiped before the transformation is 1 × 1 × 1 = 1.

Let an arbitrary three-by-three matrix, M3, be

The position vectors of the unit vectors after the transformation relative to the original coordinate system will be

Therefore, the corresponding parallelepiped, ABCDEFGH (Figure 2), has vertex coordinates:

Figure 2: The parallelepiped defined by the three unit vectors, ,î',  ĵ', and k̂'. The label of the points on the figure is concordant with their label in the article. Graph drawn using GeoGebra.

To find the volume of the parallelepiped ABCDEFGH, we first construct the cuboid with the least volume, AIJKLMNH, which contains the entirety of the parallelepiped with the three faces parallel to the x — y  , x — z , and  y — z  planes, respectively.

Figure 3: The smallest cuboid which contains the entirety of the parallelepiped (shaded yellow) with the three faces parallel to the x — y  , x — z , and  y — z  planes accordingly. Graph drawn using GeoGebra.

Volumes will be removed from this cuboid in later steps to find the volume of the parallelepiped ABCDEFGH, similar to the process in section 3.1, where a rectangle is constructed and areas are later removed from it. The volume of the cuboid is

Then, remove all excess prisms with their bases parallel to the x — y plane[2] from the cuboid to obtain a hexagonal prism[3], AOPJQRSMTUHV, so that the projection of the prism on the x — y  plane coincides with the projection of the parallelepiped on the x — y  plane (fig.4, where (a) is the shape before removing the excess prisms, and (b) is the remaining hexagonal prism after). The volume of this hexagonal prism is

Note that if the matrix represented by the parallelepiped has b = 0 or d = 0, when g = h = 0, no volume would be removed in this process; hence, the solid obtained is the original cuboid AIJKLMNH.

Figure 4: (a) The profile view of the volume removed vertically, shaded yellow; (b) The remaining volume, a hexagonal prism. Graph drawn using GeoGebra.

We now obtain a prism containing the parallelepiped of interest. Let point W be the projection of point C on the base of the prism parallel to the x — y plane (fig.5, (a)). Connect OW, RW, and JW to divide the base into three parallelograms with areas

Figure 5: (a) The volume removed from the hexagonal prism under face ABCD; (b) The quadrilateral prism with doubled the volume of the body shade blue in fig. 5.a.; (c) A half of the total volume removed from the hexagonal prism to obtain the volume of the parallelepiped, shaded blue, yellow and purple. Graph drawn using GeoGebra.

By constructing a prism with base AOWR and height (c + f), and comparing it to the space directly below the face ABCD, we find that the face ABCD of the parallelepiped divides the prism into two congruent halves. One of these halves corresponds exactly to the volume bounded by the face ABCD, the base AOWR, and lines parallel to the z-axis passing through points B, C, and D (fig.5 (a) & (b)). A similar construction can be made for two additional prisms: one with base RWJQ and height (c + 2f + i), and another with base OWJP and height (2c + f + i) (fig,5 (c)).

Since the space between the outer hexagonal prism and the inner parallelepiped is symmetric—meaning the regions bounded by faces ABO, BOPG, GPJH, HJQE, EQRD, DRA, ABCD, GBCH, HCDE, AOPJQR, and those bounded by HUG, GUTB, TBAM, MADS, SDEV, VEH, EFGH, ABGF, AFED, SMTUHV are congruent—the volume difference between the hexagonal prism and the parallelepiped is fully accounted for by the combined volume of the three constructed prisms with parallelogram bases.

Therefore, the volume of the parallelepiped is

Which is the same as the formula derived algebraically. 

If four of the vertices of a parallelepiped — A, B, C, and D — are given such that points B, C, and D are each connected to point A by an edge of the parallelepiped (as illustrated in Figure 6), and their positions are expressed relative to point A, then the coordinates of B, C, and D can be written as

The volume of the parallelepiped can be calculated by

Figure 6: (a) Points A, B, F, D form an arrangement of points that the formula is applicable when their coordinates are given; (b) Points A, B, F, H form an arrangement of points that the formula is not applicable as the edge AH passes through the parallelepiped. Graph drawn using GeoGebra.

3.3 Example Calculation

This section showcases an example of volume calculation with the determinants. This determinant method is also compared and discussed with a common method using the dot and cross products.

Suppose a parallelepiped has the vertices with coordinates

and we want to find its volume.

3.3.1 Using the determinant

Points A, B, C and H form such an arrangement mentioned above with point H being the common central point, therefore the position vectors of point A, B and C should be calculated:

Similar operations can be done to point B and C to obtain position vectors HB and HC

Let A be a three-by-three vector with entries

 The volume of this parallelepiped is then

3.3.1 Using Dot and Cross Products

For a parallelepiped with edge lengths given to be a, b and c, its volume formula can be expressed in the form with dot and cross products[6],

Both methods are suitable when the coordinates of the vertices are given, rather than when the edge lengths are given. They are of about the same computational complexity when broken down, and are both valuable ways of finding the volume of a parallelepiped. The method in section 3.3.2 is a more direct approach, where the cross product is used to find the base area and the dot product is used to find the vertical height, while the method introduced in this article approaches the volume by removing extra parts from a larger body that includes the parallelepiped. Nevertheless, both methods are less helpful when the scalar lengths of the edges and the angle between them are given, as both methods are based on the separated components of the position vectors.

4. Conclusion

This article demonstrates the formula of the determinant of a three-by-three matrix geometrically by exhibiting the isomorphism between the determinant of a three-by-three matrix, an algebraic parameter, and the volume of a parallelepiped, a geometric quantity. This perspective allows efficient volume calculation when given only the vertex coordinates, compared to geometric methods that calculate the edge lengths, base areas and height, where the calculation involves square roots and trigonometric functions. Though volume calculations are done by computers nowadays via rapid numerical approaches in practical applications, it is educationally and analytically valuable to understand the same concepts from different approaches and be aware of their applicable scenarios to allow efficient calculations.

References

[1] Francisco J. Duarte. Laser beam propagation matrices. Jan. 1, 2003, pp. 93–114. doi: 10.1016/b978-012222696-0/50053-3.

[2] S. J. Rayate. “Application Of Matrices In Engineering”. In: International Journal for Research in Engineering Application & Management 4. Special Issue-ICRTET-2018 (May 30, 2018), pp. 320–322. issn: 2454-9150.

[3] Brigham Young University, Department of Electrical and Computer Engineering. BYU Photonics - ABCD Matrix Analysis Tutorial/Ray Transfer Matrix Analysis/Transfer Matrices. url: http: //www.photonics.byu.edu/ABCD_Matrix_tut.phtml.

[4] Arfken, George B., et al. “Chapter 8 - CONSERVATION OF ENERGY.” University Physics, edited by George B. Arfken et al., Academic Press, 1984, pp. 152–72. ScienceDirect, https://doi.org/10.1016/B978-0-12-059860-1.50013-8.

[5] Dan Margalit and Joseph Rabinoff. 4.3. Determinants and Volumes. Georgia Institute of Technology, June 3, 2019, pp. 223–224.

[6] Duane Q. Nykamp. Volume of parallelepiped. url: https ://mathinsight.org/image/ volume_parallelepiped.