Dave4Math / Courses / Determinants


Learn everything about determinants from linear algebra. You can master their properties, their definition and computation, and many applications. With this course, determinants no longer have to be challenging.


January 11, 2021

Created by Dave

This course begins Summer 2021.
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Have you ever thought that math is just a big box of rules and that nothing seemed to be connected? Is it just a bunch of laws, or is there some understanding to it all that you’re missing? Introducing Dave’s online math course: Determinants. Dave consistently reminds you of previous topics and connects them to new ones when starting new material. His lessons are very prepared and planned out. It’s time to experience it all!

Who This Course is For

  • Students in college who want to learn more mathematics.
  • Anyone interested in determinants.
  • Students who are enrolled in linear algebra and want to improve their grade.
  • Anyone interested in majoring in mathematics, physics, or engineering.
  • Anyone wanting to learn about determinants and their properties, and applications of them.


There is no required textbook, though you will need an up-to-date web browser, paper, and pen. A hand-held scientific calculator of your choice is recommended.

What You’ll Learn in Determinants

  • The Determinant of a 3 by 3 Matrix
  • Linearity Properties of the Determinant
  • The Determinant of an n by n Matrix
  • The Determinant of a Block Matrix
  • The Determinant of the Transpose
  • Determinant of a Product
  • The Determinant of an Inverse
  • Minors and Laplace Expansion
  • Determinants and Gauss–Jordan Elimination
  • The Determinant of a Linear Transformation
  • The Determinant as Area and Volume
  • The Determinant as Expansion Factor
  • Cramer’s Rule
  • Adjoint and Inverse of a Matrix

Course Description

We begin this course with an introduction to 3 by 3 determinants and a discussion of Sarrus’s rule. We then illustrate several properties of the determinant before examining the definition of an n by n determinant using patterns and inversions. After this vital definition, we work through several examples, including the determinant of a triangular matrix and determinants of block matrices.

Once it is clearly understood what a determinant is, in the next lesson, we dive deep into the determinant’s properties. The linearity properties of the determinant and finding determinants using Gaussian elimination are examined. The determinant of products, powers, inverses, and similar matrices are also covered. Following this, we consider the determinant of a linear transformation.

In the final lesson, we study several special cases; for example, an orthogonal matrix’s determinant is either 1 or -1. We also explore the geometrical interpretations of the determinants, including areas and volumes. At the end of this lesson, we discuss the determinant as an expansion factor with linear transformations.

In conclusion, we motivate, prove, and demonstrate Cramer’s rule in solving systems of linear equations.

Recommended Prerequisites for Determinants

I recommend the prerequisite course Orthogonality. If you’re not sure if this course is for you, checkout the course contents below or find out more by checking out my free articles on linear algebra.

Course Content

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David A. Smith

Mathematics Educator

I sincerely believe that the potential in every student can be unlocked. As an accomplished and dedicated instructor, I have firm confidence that I can provide significant value to your studies.

I relish teaching and use it as a channel for my creativity as I seek to deliver real-life and relevant connections between the mathematics that I teach and my students. Promoting an active learning environment and excitement about mathematics is my dream come true.

David Smith (Dave) has a B.S. and M.S. in Mathematics and has enjoyed teaching precalculus, calculus, linear algebra, and number theory at both the junior college and university levels for over 20 years. David is the founder and CEO of Dave4Math.