##### Objectives
1. Understand what it means for a square matrix to be invertible.
2. Learn about invertible transformations, and understand the relationship between invertible matrices and invertible transformations.
3. Recipes: compute the inverse matrix, solve a linear system by taking inverses.
4. Picture: the inverse of a transformation.
5. Vocabulary words: inverse matrix, inverse transformation.

In Section 3.1 we learned to multiply matrices together. In this section, we learn to “divide” by a matrix. This allows us to solve the matrix equation in an elegant way:

One has to take care when “dividing by matrices”, however, because not every matrix has an inverse, and the order of matrix multiplication is important.

# Subsection3.5.1Invertible Matrices

The reciprocal or inverse of a nonzero number is the number which is characterized by the property that For instance, the inverse of is We use this formulation to define the inverse of a matrix.

##### Definition

Let be an (square) matrix. We say that is invertible if there is an matrix such that

In this case, the matrix is called the inverse of and we write

We have to require and because in general matrix multiplication is not commutative. However, we will show in this corollary in Section 3.6 that if and are matrices such that then automatically

##### Proof

1. The equations and at the same time exhibit as the inverse of and as the inverse of
2. We compute
Here we used the associativity of matrix multiplication and the fact that This shows that is the inverse of

Why is the inverse of not equal to If it were, then we would have

But there is no reason for to equal the identity matrix: one cannot switch the order of and so there is nothing to cancel in this expression. In fact, if then we can multiply both sides on the right by to conclude that In other words, if and only if

More generally, the inverse of a product of several invertible matrices is the product of the inverses, in the opposite order; the proof is the same. For instance,

# Subsection3.5.2Computing the Inverse Matrix¶ permalink

So far we have defined the inverse matrix without giving any strategy for computing it. We do so now, beginning with the special case of matrices. Then we will give a recipe for the case.

##### Definition

The determinant of a matrix is the number

There is an analogous formula for the inverse of an matrix, but it is not as simple, and it is computationally intensive. The interested reader can find it in this subsection in Section 4.2.

The following theorem gives a procedure for computing in general.

# Subsection3.5.3Solving Linear Systems using Inverses

In this subsection, we learn to solve by “dividing by

##### Proof

The advantage of solving a linear system using inverses is that it becomes much faster to solve the matrix equation for other, or even unknown, values of For instance, in the above example, the solution of the system of equations

where are unknowns, is

As with matrix multiplication, it is helpful to understand matrix inversion as an operation on linear transformations. Recall that the identity transformation on is denoted

##### Definition

A transformation is invertible if there exists a transformation such that and In this case, the transformation is called the inverse of and we write

The inverse of “undoes” whatever did. We have

for all vectors This means that if you apply to then you apply you get the vector back, and likewise in the other order.

As you might expect, the matrix for the inverse of a linear transformation is the inverse of the matrix for the transformation, as the following theorem asserts.