Overview
On this page, we discuss applications of determinants, including an alternative method to find the inverse of a matrix and Cramer’s Rule.
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.
 Compute the matrix of cofactors for a square matrix (review).
 Compute the adjoint matrix, \(\text{adj}(A)\), for a square matrix \(A\).
 Use the adjoint of an invertible matrix to find its inverse.
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 Cramer’s rule to solve linear systems of equations.
To prepare for class

Watch this video by patrickJMT which shows how to use cofactors to find the inverse of a \(3\times 3\) matrix:

Watch this video by patrickJMT which shows how to use Cramer’s Rule to solve a system of 3 linear equations in 3 unknowns:
After class

(Optional) Watch this video by 3Blue1Brown which explains the geometric idea behind Cramer’s Rule: