Overview
On this page, we discuss properties of determinants, and in particular how they are affected by row operations.
Important
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.
- Know the effect of elementary row operations on the value of a determinant.
- Know the determinant of the three types of elementary matrices.
- Know how to introduce zeros in the rows or columns of a matrix to facilitate the evaluation of its determinant (i.e. which row/column operation(s) to apply).
- Know how det(\(A)\) and det(\(A^{-1}\)) are related.
- Know how determinants behave with respect to basic arithmetic operations (scalar multiplication, addition, matrix multiplication).
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 row reduction to evaluate the determinant of a matrix.
- Use column operations to evaluate the determinant of a matrix.
- Combine the use of row reduction and cofactor expansion to evaluate the determinant of a matrix.
To prepare for class
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Watch this great video by 3Blue1Brown which explains the relationship between a linear transformation and the determinant of its matrix and how this helps us better understand properties of determinants:
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Watch this video by TheTrevTutor which explains how Row Operations affect the determinant of a matrix, as well as some other important properties of determinants:
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Watch this video by Khan Academy which shows how to use row operations to simplify the calculation of the determinant of a \(4\times 4\) matrix:
