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Section4.5Matrix Inverses

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. Theorem: the invertible matrix theorem.
  6. Vocabulary words: inverse matrix, inverse transformation.

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 Ax = b in an elegant way:

Ax = b ⇐⇒ x = A 1 b .

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.

Subsection4.5.1Invertible Matrices

The reciprocal or inverse of a nonzero number a is the number b which is characterized by the property that ab = 1. For instance, the inverse of 7 is 1 / 7. We use this formulation to define the inverse of a matrix.

Definition

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

AB = I n and BA = I n .

In this case, the matrix B is called the inverse of A , and we write B = A 1 .

We have to require AB = I n and BA = I n because in general, matrix multiplication is not commutative. However, we will show in the invertible matrix theorem that if A and B are n × n matrices such that AB = I n , then automatically BA = I n .

Proof

  1. The equations AA 1 = I n and A 1 A = I n at the same time exhibit A 1 as the inverse of A and A as the inverse of A 1 .
  2. We compute
    ( B 1 A 1 ) AB = B 1 ( A 1 A ) B = B 1 I n B = B 1 B = I n .
    Here we used the associativity of matrix multiplication and the fact that I n B = B . This shows that B 1 A 1 is the inverse of AB .

Why is the inverse of AB not equal to A 1 B 1 ? If it were, then we would have

I n =( AB )( A 1 B 1 )= ABA 1 B 1 .

But there is no reason for ABA 1 B 1 to equal the identity matrix: one cannot switch the order of A 1 and B , so there is nothing to cancel in this expression.

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,

( ABC ) 1 = C 1 B 1 A 1 .

Subsection4.5.2Computing the Inverse Matrix

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 2 × 2 matrices.

Definition

The determinant of a 2 × 2 matrix is the number

det F abcd G = ad bc .

There is an analogous formula for the inverse of an n × n matrix, but it is not as simple, and it is computationally intensive. The interested reader can find it in subsection in Section 5.2.

The following theorem gives a procedure for computing A 1 in general.

Proof

Subsection4.5.3Solving Linear Systems using Inverses

In this subsection, we learn to solve Ax = b by “dividing by A .

Proof

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

E 2 x 1 + 3 x 2 + 2 x 3 = b 1 x 1 + 3 x 3 = b 2 2 x 1 + 2 x 2 + 3 x 3 = b 3 ,

where b 1 , b 2 , b 3 are unknowns, is

C x 1 x 2 x 3 D = C 232103223 D 1 C b 1 b 2 b 3 D = C 6 5932 422 3 DC b 1 b 2 b 3 D = C 6 b 1 5 b 2 + 9 b 3 3 b 1 + 2 b 2 4 b 3 2 b 1 + 2 b 2 3 b 3 D .

Subsection4.5.4Invertible linear transformations

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

Definition

A transformation T : R n R n is invertible if there exists a transformation U : R n R n such that T U = Id R n and U T = Id R n . In this case, the transformation U is called the inverse of T , and we write U = T 1 .

The inverse U of T “undoes” whatever T did. We have

T U ( x )= x and U T ( x )= x

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

Proof

Subsection4.5.5Characterizing Invertible Matrices

This subsection consists of a single important theorem, with many equivalent conditions for a matrix to be invertible. This is one of the most important theorems in this textbook. We will append two more criteria in Section 6.1.

Proof

There are two kinds of square matrices:

  1. invertible matrices, and
  2. non-invertible matrices.

For invertible matrices, all of the statements of the invertible matrix theorem are true.

For non-invertible matrices, all of the statements of the invertible matrix theorem are false.