1

How do you solve the equation  5x = 10 ?

We solve a matrix equation similarly:    AX = B

So how do we find A­1 and what does "1" look like in matrices?

2

6.3 Inverses of Matrices & Matrix Equations

IDENTITY MATRIX

The identity matrix In is the n n matrix for which each main diagonal entry is a 1 and for which all other entries are 0.

An Identity Matrix must be a  

square matrix!

3

Multiply the following Matrices:

=

4

INVERSE OF A MATRIXLet A be a n   n matrix. If there exists an n   n matrix A­1 with the property that

Then we say that A­1 is the inverse of A

5

Verify that B is the inverse of A, where

B

6

INVERSE OF A 2 X 2 MATRIX

If ,then

If ad ­ bc=0 then A has no inverse.

So now we know how to verify if two matrices are inverses,

 but how do we find an inverse matrix?

We call ad ­ bc  the 

"determinant"

7

a) Let

If possible, find A­1, and verify that

b) Let B 6

8

To find the inverse of an n x n matrix --> Augment the matrix with the identity matrix In of the same size on the right:

Then use elementary row operations to change the right side into the identity matrix. The right n x n piece is then A­1.

In other words if and Then

Only Matrices that are square have inverses!

9

Let A be the matrix

Find A­1  and then verify that it is the inverse.

10

Find the inverse for

11

If we encounter a row of zeros on the left when trying to find an inverse, then the original matrix does not have an inverse.

12

SOLVING A MATRIX EQUATION

If A is a square n x n matrix that has an inverse A­1 and if X is a variable matrix and B a known matrix, both with n rows, then the solution of the matrix equation

is given by

13

Solve the following system using a matrix equation(The inverse work has already been done on a previous example.)

14

Solve the following system using a matrix equation

15

Assignment:

Section 6.3: problems 1­7 odd, 11­25 odd, 39­45 odd, 53

For problems 39­45, the inverses of each coefficient matrix is from previous problems: #11, 13, 19, & 23, so 

you don't have to do the inverse work twice. 

Also, let's look at #53 together before you start your homework.

So just find A­1 and multiply it by B, just we did in our previous examples.