What invertible means
A square matrix is invertible if there is another matrix that undoes it.
If A is invertible, then A^{-1} exists and
Definition
Invertible matrix
An invertible matrix has a matrix inverse, so multiplying by that inverse recovers the identity matrix.
How to think about it
Invertibility is not just a symbol game. It tells you whether the matrix has a reversible effect. That is why row reduction is so useful: it lets you test whether the matrix can be turned into the identity matrix.
Worked example
Read invertibility from row reduction
Suppose row operations transform A into I.
Then the same operations applied to [A | I] produce [I | A^{-1}].
This is the practical reason we keep row-reducing augmented blocks in the later pages of the course.
Step through [A | I] below and watch what you are really waiting for. The key
moment is not “many operations happened,” but “the left block became I.”
Try it here
Invertibility by row reduction
The live demo lets you step through [A | I] until the left block becomes I.
Start from [A | I]. If A is invertible, row reduction will turn the left block into I.
| 1 | 2 | 1 | 1 | 0 | 0 |
| 0 | 1 | 1 | 0 | 1 | 0 |
| 2 | 3 | 4 | 0 | 0 | 1 |
Common mistake
Common mistake
Simple-looking is not the same as invertible
Do not guess from appearance alone. The real test is whether the matrix can be reduced to the identity.
Quick check
Quick check
If `A` is invertible, what is `AA^{-1}`?
Solution
Answer
Prerequisite link
This page depends on 2.3 Gaussian elimination and RREF.