Overview
On this page, we discuss elementary matrices and their role in finding the inverse of an arbitrarily large matrix.
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.
- Determine whether a given square matrix is elementary.
- Determine the elementary row operation corresponding to a given elementary matrix.
- Write the elementary matrix corresponding to a given elementary row operation.
- Apply elementary row operations to reduce a given square matrix to the identity matrix (review).
- Apply the inverse of a given elementary row operation to a matrix.
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 the inversion algorithm to find the inverse of an invertible matrix.
- Understand various statements that are equivalent to the invertibility of a square matrix.
- Express an invertible matrix as a product of elementary matrices.
- Be able to prove basic properties involving row operations and row-equivalent matrices.
- Determine whether two square matrices are row-equivalent.
- Determine whether a diagonal matrix is invertible with no computations.
To prepare for class
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Watch this video which introduces elementary matrices and shows how they give us an efficient method of finding the inverse of any invertible \(n\times n\) matrix (Note: the first matrix \(E_1\) shown at 0:25 in the video should have \(+4\) in the bottom left entry instead of \(-4\)):
After class
The ideas behind the inversion algorithm in fact show an even more useful general statement:
Theorem:
If row-reducing the augmented matrix \([A|I]\) (where \(I\) is an identity matrix of appropriate size) results in the matrix \([B|P]\), then we have \(B=PA\). In other words, the matrix \(P\) can be thought of as the transformation matrix which has the same effect as the sequence of row operations which transforms \(A\) into \(B\).
As a special case, if \([A|I]\) row-reduces to \([I|P]\), then we have \(I=PA\), which means that \(P\) is in fact equal to \(A^{-1}\), the inverse of \(A\).
Note that \(P\) is in fact equal to the product of the elementary matrices corresponding to the row operations applied to \(A\).
