In Section 4.1 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.
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.
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 4.6 that if and are matrices such that then automatically
Let and be invertible matrices.
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,
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.
The determinant of a matrix is the number
Let
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 5.2.
The following theorem gives a procedure for computing in general.
Let be an matrix, and let be the matrix obtained by augmenting by the identity matrix. If the reduced row echelon form of has the form then is invertible and Otherwise, is not invertible.
In this subsection, we learn to solve by “dividing by ”
Let be an invertible matrix, and let be a vector in Then the matrix equation has exactly one solution:
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
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.
Let be a linear transformation with standard matrix Then is invertible if and only if is invertible, in which case is linear with standard matrix
