The inverse of a matrix A is a matrix that, when multiplied by A results in the identity. The notation for this inverse matrix is A–1.
You are already familiar with this concept, even if you don’t realize it! When working with numbers such as 3 or –5, there is a number called the multiplicative inverse that you can multiply each of these by to get the identity 1. In the case of 3, that inverse is 1/3, and in the case of –5, it is –1/5.
Does every matrix have an inverse?
Thinking about the number 0, there is no number you can multiply it by to get 1. So, the number 0 has no multiplicative inverse.
Similarly, not every matrix has an inverse. For it to even be a possibility, the matrix must first be square (same number of rows as columns). Even then, there may not be an inverse. When talking about a matrix with or without an inverse, the following terminology is used:
- A matrix is said to be invertible or, less commonly, nonsingular if it has an inverse.
- A matrix is said to be singular or not invertible if it does not have an inverse.
Often, you can’t simply look at a matrix and tell whether it is invertible or not. Consider the following matrix.
You can verify that this matrix not invertible using your calculator. Or, ff you have studied a lot of linear algebra, you may be able to tell by carefully inspecting the columns (hint: it has to do with linear dependence).
How can we determine if a matrix is invertible?
This is one of the biggest areas of study in a linear algebra course, since, it turns out that invertible matrices have connections back to systems of equations and to other concepts like linear independence or dependence. This idea will be explored in future articles.