On this page, we discuss properties of matrix arithmetic operations, as well as the existence (or not) of the inverse of a matrix and its consequences.
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.
- Know the arithmetic properties of matrix operations.
- Know the properties of zero matrices.
- Know the properties of identity matrices.
- Be able to recognize when two square matrices (whose entries are given) are inverses of each other.
- Be able to determine whether a \(2 \times 2\) matrix is invertible.
- Compute the inverse of an invertible \(2 \times 2\) matrix using the formula.
- Know the properties of the matrix transpose and its relationship with invertible matrices.
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
- Be able to prove arithmetic properties of matrices.
- Be able to recognize when two square matrices are inverses of each other.
- Be able to prove basic properties involving invertible matrices.
To prepare for class
Watch this short video which shows how to quickly find the inverse of a \(2\times 2\) matrix:
Watch this short video which shows how to use the inverse (if it exists) of the \(2\times 2\) coefficient matrix of a SLE to quickly find the unique solution: