Definition
For a square matrix, i.e., a matrix with the same number of rows and columns, one can capture important information about the matrix in a just single number, called the determinant. The determinant is useful for solving linear equations, capturing how linear transformation change area or volume, and changing variables in integrals.6
Who
For those who learned about matrix determinants in a linear algebra course and presentation of the material may have been very dry and mathematical. So you do not understand the usefulness of determinants. You are not alone. Here is what some have said about learning linear algebra:
Despite two linear algebra classes, my knowledge consisted of “Matrices, determinants, eigen something something”.2
What
In mathematics, how to compute something should never be the first question. The first question is always: “What really IS it?” Only then should we ask: “Ok now that we know what it is, how can we compute it.”
The determinant of a matrix is the factor by which areas are scaled by this matrix.
However, regarding matrix determinants, I was taught that they are numbers for matrices, how to compute them, and not much more. It took until my university courses that I learned the beauty behind determinants.
As soon as I learned about determinants’ geometric meaning, I was wondering why this wasn’t already taught in high school as it is very easy to understand and mind-opening.1
The determinant is the “size” of the output transformation. If the input was a unit vector (representing area or volume of 1), the determinant is the size of the transformed area or volume. A determinant of 0 means matrix is “destructive” and cannot be reversed (similar to multiplying by zero: information was lost).2
Whatever area in the input space we choose, it seems that after the transformation the area gets bigger. This is precisely what the determinant is!
The determinant of a matrix is the factor by which areas are scaled by this matrix.
Because matrices are linear transformations it is enough to know the scaling factor for one single area to know the scaling factor for all areas.1
Why
See Theoretical Knowledge Vs Practical Application.
How
I don’t show you how to to compute eigenvalues and eigenvectors. Many of the References, Additional Reading, websites and YouTube videos will assist you with that. As some professors say: “It is intuitively obvious to even the most casual observer.”
Cramer’s Rule
The Cramer’s rule provides a method of solving a system of linear equations through the use
of determinants. The Hessian4 5 can be used for a test whether a given critical point is a local
minimum or maximum of a function of two or more real variables. In the article the use of the
Cramer’s rule and the Hessian is demonstrated on economic optimization problems.
Determinants and Cramer’s rule are important tools for solving many problems in business
and economy. Especially for searching an optimal solution of the maximization profit or
minimization cost problems it can be very often apply.3
Applying Cramer’s Rule
I try not to provide examples unless I think they are needed. In this case, this example shows the usefulness of matrices and Cramer’s Rule.7 8
Cramer’s rule is a way of solving a system of linear equations using determinants. Suppose we are trying to solve a system of linear equations such as the following.
or Ax = b in matrix form, where
To find the value of each variable, you divide the determinant of each corresponding modified matrix by the determinant of the coefficient matrix.
For our example, the determinant of the coefficient matrix A is as follows.
The corresponding modified matrix Ai is a new matrix formed by replacing the ith column of A with the b vector.
Using Cramer’s rule and doing the dividing.
So the solution to the system of equations is x = 3, y = -2, z = -1.
Limitations of Cramer’s rule
- Because we are dividing by det(A) to get
, Cramer’s rule only works if det(A) ≠ 0. If det(A) = 0, Cramer’s rule cannot be used because a unique solution does not exist since there would be infinitely many solutions, or no solution at all. - Cramer’s rule is slow because we have to evaluate a determinant for each xi. When we evaluate each det(Ai) we have to perform Gaussian elimination on each Ai for a total of n times. In comparison, if we were to use the augmented matrix, [A|b], we would only need to perform Gaussian elimination once to solve Ax = b.
Application
“Applications Of Matrices And Determinants – Geeksforgeeks”. 2021. geeksforgeeks. https://www.geeksforgeeks.org/applications-of-matrices-and-determinants/.
“6.5 – Applications Of Matrices And Determinants”. 2021. people.richland.edu. https://people.richland.edu/james/lecture/m116/matrices/applications.html.
References
1 “What Really IS A Matrix Determinant?” 2021. Medium. https://towardsdatascience.com/what-really-is-a-matrix-determinant-89c09884164c.
2 “An Intuitive Guide to Linear Algebra”. 2021. betterexplained.com. https://betterexplained.com/articles/linear-algebra-guide/.
3 Valentová, Eva. DETERMINANTS AND THEIR USE IN ECONOMICS. 2021. opf.slu.cz. http://www.opf.slu.cz/aak/2011/03/valentova.pdf.
4 “The Hessian Matrix | Multivariable Calculus (Article) | Khan Academy”. 2021. Khan Academy. https://www.khanacademy.org/math/multivariable-calculus/applications-of-multivariable-derivatives/quadratic-approximations/a/the-hessian.
The Hessian is a matrix that organizes all the second partial derivatives of a function.
5 “Hessian Matrix | Brilliant Math & Science Wiki”. 2021. brilliant.org. https://brilliant.org/wiki/hessian-matrix/.
The Hessian Matrix is a square matrix of second ordered partial derivatives of a scalar function. It is of immense use in linear algebra as well as for determining points of local maxima or minima.
6 “The Determinant Of A Matrix – Math Insight”. 2021. mathinsight.org. https://mathinsight.org/determinant_matrix.
7 Sterling, Mary Jane. Linear Algebra for Dummies. Hoboken, NJ: Wiley Publishing, Inc., 2009.
8 “Cramer’s rule”. 2022. Math.net. https://www.math.net/cramers-rule.
Additional Reading
“Determinants: Definition”. 2021. textbooks.math.gatech.edu. https://textbooks.math.gatech.edu/ila/determinants-definitions-properties.html.
“Understanding Properties Of Determinants | College Algebra”. 2021. courses.lumenlearning.com. https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/understanding-properties-of-determinants/.
V, Uma. 2021. “Applications Of Determinants And Matrices: Meaning, Solved Examples”. Embibe Exams. https://www.embibe.com/exams/applications-of-determinants-and-matrices/.
Mathematical Mysteries 