4.7 Identity and Inverse Matrices

-Identity matrices-Inverse matrix (intro)-An application-Finding inverse matrices (by hand)-Finding inverse matrices (using calculator)

A review of the Identity

• For real numbers, what is the additive identity?

• Zero…. Why?

• Because for any real number b, 0 + b = b

• What is the multiplicative identity?

• 1 … Why?

• Because for any real number b, 1 * b = b

Identity Matrices

• The identity matrix is a square matrix (same # of rows and columns) that, when multiplied by another matrix, equals that same matrix

• If A is any n x n matrix and I is the n x n Identity matrix, then A * I = A and I*A = A

Examples

• The 2 x 2 Identity matrix is:

• The 3 x 3 Identity matrix is:

1 0

0 1

1 0 0

0 1 0

0 0 1

•Notice any pattern?

•Most of the elements are 0, except those in the diagonal from upper left to lower right, in which every element is 1!

Inverse review

• Recall that we defined the inverse of a real number b to be a real number a such that a and b combined to form the identity

• For example, 3 and -3 are additive inverses since 3 + -3 = 0, the additive identity

• Also, -2 and – ½ are multiplicative inverses since (-2) *(- ½ ) = 1, the multiplicative identity

Matrix Inverses

• Two n x n matrices are inverses of each other if their product is the identity

• Not all matrices have inverses (more on this later)

• Often we symbolize the inverse of a matrix by writing it with an exponent of (-1)

• For example, the inverse of matrix A is A-1

• A * A-1 = I, the identity matrix.. Also A-1 *A = I• To determine if 2 matrices are inverses, multiply

them and see if the result is the Identity matrix!

Determine whether X and Y are inverses.

Check to see if X • Y = I.

Write an equation.

Matrixmultiplication

Now find Y • X.

Matrixmultiplication

Write an equation.

Answer: Since X • Y = Y • X = I, X and Y are inverses.

Determine whether P and Q are inverses.

Check to see if P • Q = I.

Write anequation.

Matrix multiplication

Answer: Since P • Q I, they are not inverses.

Determine whether each pair of matrices are inverses.

a.

b.

Answer: no

Answer: yes

An Application of Inverse Matrices

• You can use matrices to encode and decode a message• In other words, matrices are useful for encrypting

information• First, translate your message into numbers using the key

A = 1, B = 2, etc.. (perhaps 0 = space)• Organize your message into a matrix with 2 columns and

as many rows as needed• Multiply the matrix by a 2 x 2 encoding matrix• To decipher the message, multiply the coded message

by a 2 x 2 decoding matrix• The decoding matrix will be the inverse of the encoding

matrix• Finally, you can translate the numbers back into letters

using you’re the key mentioned above

Use the table to assign a number to each letter in the message ALWAYS_SMILE. Then code the message with

the matrix

Code

_ 0 I 9 R 18

A 1 J 10 S 19

B 2 K 11 T 20

C 3 L 12 U 21

D 4 M 13 V 22

E 5 N 14 W23

F 6 O 15 X 24

G 7 P 16 Y 25

H 8 Q 17 Z 26

A L W A Y S _ S M I L E

1 12 23 1 25 19 0 19 13 9 12 5

Convert the message to numbers using the table.

Write the message in matrix form. Then multiply the message matrix B by the coding matrix A.

Write an equation.

Matrix multiplication

Simplify.

Answer: The coded message is 13 | 38 | 24 | 49 | 44 | 107 | 19 | 57 | 22 | 53 | 17 | 39.

Now decode the message

13 | 38 | 24 | 49 | 44 | 107 | 19 | 57 | 22 | 53 | 17 | 39

• Decode by:

• expressing the coded message as a matrix with 2 columns

• Multiplying this matrix by the inverse of A

• The inverse of A is shown below:

3 2

1 1

Next, decode the message by multiplying the coded matrix C by A–1.

Use the table again to convert the numbers to letters. You can now read the message.

Code

_ 0 I 9 R 18

A 1 J 10 S 19

B 2 K 11 T 20

C 3 L 12 U 21

D 4 M 13 V 22

E 5 N 14 W23

F 6 O 15 X 24

G 7 P 16 Y 25

H 8 Q 17 Z 26Answer:

1 12 23 1 25 19 0 19 13 9 12 5

A L W A Y S _ S M I L E

Code

_ 0 I 9 R 18

A 1 J 10 S 19

B 2 K 11 T 20

C 3 L 12 U 21

D 4 M 13 V 22

E 5 N 14 W23

F 6 O 15 X 24

G 7 P 16 Y 25

H 8 Q 17 Z 26

a. Use the table to assign a number to each letter in the message FUN_MATH. Then code the message

with the matrix A =

Answer: 12 | 63 | 28 | 14 | 26 | 16 | 40 | 44

Example 7-3k

Code

_ 0 I 9 R 18

A 1 J 10 S 19

B 2 K 11 T 20

C 3 L 12 U 21

D 4 M 13 V 22

E 5 N 14 W23

F 6 O 15 X 24

G 7 P 16 Y 25

H 8 Q 17 Z 26

Answer:

6 21 14 0 13 1 20 8

F U N _ M A T H

Use the inverse matrix shown below to decode the message!!

1 12 6

10 3

How do we find the inverse???

• 1st you find what is called the determinant• The determinant not only determines whether the

inverse of a matrix exists, but also influences what elements the inverse contains

• For the matrix shown below, the determinant is equal to ad – bc

• In other words, multiply the elements in each diagonal, then subtract the products!

a b

c d

More about determinants

• If the determinant of a matrix equals zero, then the inverse of that matrix does not exist!

• Every square matrix has a determinant, however 2 x 2 matrices are the only ones we will calculate determinants for by hand

• For larger matrices, finding the determinant is considerably more complicated (if you take a linear programming course in college, or AP Physics here at WHS, you may learn how to find 3 x 3 determinants by hand)

Finding the inverse of a 2 x 2 matrix

• For the matrix:

• The inverse is found by calculating:

a b

c d

1 d b

c aad bc

In other words: -Switch the elements a and d -Reverse the signs of the elements b and c -Multiply by the fraction (1 / determinant)

Find the inverse of the matrix, if it exists.

Find the value of the determinant.

Since the determinant is not equal to 0, S –1 exists.

Definition of inverse

a = –1, b = 0,c = 8, d = –2

Answer: Simplify.

Check:

Find the inverse of the matrix, if it exists.

Find the value of the determinant.

Answer: Since the determinant equals 0, T –1 does

not exist.

Find the inverse of each matrix, if it exists.

a.

b.

Answer: No inverse exists.

Answer:

Finding inverses for larger matrices

• We will not calculate inverses of 3 x 3 or larger matrices by hand

• However, we CAN find these using the TI-83• Enter your matrix using the EDIT menu, then

print it on your TI screen using the NAMES menu

• Now hit the “X-1” button to indicate that you want to find the inverse of this matrix!

• Let’s try some examples on the TI-83!!