On this page, we discuss methods to find Determinants of arbitrarily large square matrices using the method of cofactor expansions (sometimes also called Laplace expansion).
Note: The basic and advanced learning objectives listed below are meant to give you an idea of the material you should learn about this section. These are mainly intended to be used in a course which uses an Active Learning approach, where students are required to "read ahead" before each class - but can equally be used in a more traditional course setting.
Unless your teacher gives you specific instructions, it is up to you to decide how much of the listed resources you need to read or watch - you probably do not need to go through all of it. You might also want to look at the General Study Tips & Tricks page for some recommendations on how to effectively study with a math textbook and videos.
Basic learning objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.
- Find the determinant of a \(2\times 2\) or \(3\times 3\) matrix using the "basketweave" formulas.
- Use the determinant of a \(2 \times 2\) invertible matrix to find the inverse of that matrix (review).
- Find the minors and cofactors of a square matrix.
- Find the determinant of a triangular, or diagonal matrix by inspection.
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
- Use cofactor expansion to evaluate the determinant of a square matrix.
To prepare for class
Watch this short video which reviews/explains the simple formulas for finding the determinant of a \(2\times 2\) or \(3\times 3\) matrix (but note that this does not work for larger matrices):
Watch this video by TheTrevTutor which shows how to use cofactor expansion to calculate the determinant of \(3\times 3\) and \(4\times 4\) matrices, and which shows how this gives a very simple formula for a triangular matrix: