Inverse of a Matrix
Inverse Transformation
Given a transformation such that for any in i.e. does "nothing". Then by definition, it is obviously a linear transformation and we call it an identity transformation, which is usually denoted by . And the matrix for an identity transformation is called an identity matrix, denoted by (or just if there is no ambiguity). Explicitly, we have
(Note: It is an n x n matrix if the domain and range of is .)
In general, a matrix is called a square matrix if its number of rows equals its number of columns. A square matrix is called a diagonal matrix if its entries outside the main diagonal are zero. Therefore, an identity matrix is always a diagonal matrix.
Given any linear transformation . We would like to know if there is a transformation that "reverses" the action of . If it exists, we call it the inverse transformation of S, denoted by . It can be shown that is again a linear transformation (Why?) and .
Suppose is the n x n matrix for the linear transformation . Then we denote the matrix for its inverse transformation by . Since the composition of linear transformations corresponds to the matrix multiplication, we have
From the viewpoint of matrix algebra, the inverse of a matrix plays a similar role as "reciprocal" for real numbers. You can see the analogy between "" and "". When you solve a simple equation like "", you can immediately write down the solution . But the underlying procedure to obtain such a solution goes like this:
Recall that a system of n linear equation with n variables can be expressed as a matrix equation , where is the n x n matrix containing all the coefficients in the system. We can use the same idea to solve for the solution if exists:
Therefore, solving such a system of linear equations boils down to finding the inverse of A. However, this works only when exists. Usually, we say is invertible (or nonsingular) if exists. Then is not invertible (or singular) if does not exist. Even if is invertible, finding its inverse is usually not an easy task, especially when the size of the matrix is large. We will talk about the methods for computing inverses later in the course.
Inverse of an Invertible 2 x 2 Matrix
Although it is not easy to compute the inverse of a general matrix, we do have a simple formula for the inverse of an invertible 2 x 2 matrix. The following is the main result:
Theorem: Let . If , then is invertible and
If , then is not invertible.
Proof: We can do the direct verification to show the formula works. As for the case when , what can you say about the column vectors and ?
In the above theorem, we have not only a formula for the inverse of a 2 x 2 matrix, but also a condition for checking either a 2 x 2 matrix is invertible or not by checking if is non-zero. This important quantity is called the determinant of , denoted by . Determinants are so important that we will devote a whole chapter to them later.
Suppose . Find .
Using the result of the previous question, solve the following system of linear equation:
Suppose and are two non-zero square matrices of the same size such that . Prove that and must be singular.