Inverse of a Matrix

Definition and Examples

Recall that functions f and g are inverses if

        f(g(x))  =  g(f(x))  =  x

We will see later that matrices can be considered as functions from Rn to Rm and that matrix multiplication is composition of these functions.  With this knowledge, we have the following:

Let A and B be n x n matrices then A and B are inverses of each other, then

       AB  =  BA  =  In



Consider the matrices


We can check that when we multiply A and B in either order we get the identity matrix.  (Check this.)

Not all square matrices have inverses.  If a matrix has an inverse, we call it nonsingular or invertible.  Otherwise it is called singular.  We will see in the next section how to determine if a matrix is singular or nonsingular.

Properties of Inverses

Below are four properties of inverses. 

  1. If A is nonsingular, then so is A-1 and

    (A-1) -1  =  A

  2. If A and B are nonsingular matrices, then AB is nonsingular and

            (AB) -1  =  B-1A-1

  3. If A is nonsingular then

              (AT) -1  =  (A -1)T

  4. If A and B are matrices with

            AB  =  In

    A and B are inverses of each other.

Notice that the fourth property implies that if AB  =  I then BA  =  I.

The first three properties' proof are elementary, while the fourth is too advanced for this discussion.  We will prove the second.

Proof that (AB) -1  =  B -1 A -1

By property 4, we only need to show that

        (AB)(B -1 A -1)  =  I

We have

        (AB)(B -1 A -1)  =  A(BB -1)A -1     associative property

       =  AIA-1        definition of inverse

        =  AA-1       definition of the identity matrix

       =  I               definition of inverse

Finding the Inverse

Now that we understand what an inverse is, we would like to find a way to calculate and inverse of a nonsingular matrix.  We use the definitions of the inverse and matrix multiplication.  Let A be a nonsingular matrix and B be its inverse.  Then

        AB  =  I

Recall that we find the jth column of the product by multiplying A by the jth column of B.  Now for some notation.  Let ej be the m x 1 matrix that is the jth column of the identity matrix and xj be the jth column of B.  Then 

        Axj  =  ej 

We can write this in augmented form


Instead of solving these augmented problems one at a time using row operations, we can solve them simultaneously.  We solve

        [A | I]



Find the inverse of the matrix





The inverse matrix is just the right hand side of the final augmented matrix


This example demonstrates that if A is row equivalent to the identity matrix then A is nonsingular.

Linear Systems and Inverses

We can use the inverse of a matrix to solve linear systems.  Suppose that

        Ax  =  b

Then just as we divide by a coefficient to isolate x, we can apply A-1 to both sides to isolate the x

        A-1Ax  =  A-1b

        Ix  =  A-1b        x  =  A-1b



        x + 4z  =  2
        x + y + 6z  =  3
        -3x - 10z  =  4


We put this system in matrix form

       Ax  =  b



The solution is

        x  =  A-1 b

We have already computed the inverse.  We arrive at


The solution is

        x  =  -18        y  =  -9        z  =  5

Notice that if b is the zero vector, then

        Ax  =  0

can be solved by

        x  =  A-10  =  0

This demonstrates a theorem

Theorem of Nonsingular Equivalences

The Following Are Equivalent (TFAE)

  1. A is nonsingular

  2. Ax  =  0 has only the trivial solution

  3. A is row equivalent to I

  4. The linear system Ax  =  b has a unique solution for every n x 1 matrix b

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