4.7 identity and inverse matrices
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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? - PowerPoint PPT PresentationTRANSCRIPT
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4.7 Identity and Inverse Matrices
-Identity matrices-Inverse matrix (intro)-An application-Finding inverse matrices (by hand)-Finding inverse matrices (using calculator)
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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
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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
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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!
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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
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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!
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Determine whether X and Y are inverses.
Check to see if X • Y = I.
Write an equation.
Matrixmultiplication
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Now find Y • X.
Matrixmultiplication
Write an equation.
Answer: Since X • Y = Y • X = I, X and Y are inverses.
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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.
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Determine whether each pair of matrices are inverses.
a.
b.
Answer: no
Answer: yes
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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
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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.
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Write the message in matrix form. Then multiply the message matrix B by the coding matrix A.
Write an equation.
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Matrix multiplication
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Simplify.
Answer: The coded message is 13 | 38 | 24 | 49 | 44 | 107 | 19 | 57 | 22 | 53 | 17 | 39.
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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
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Next, decode the message by multiplying the coded matrix C by A–1.
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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
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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
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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
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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
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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)
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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)
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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.
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Definition of inverse
a = –1, b = 0,c = 8, d = –2
Answer: Simplify.
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Check:
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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.
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Find the inverse of each matrix, if it exists.
a.
b.
Answer: No inverse exists.
Answer:
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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!!
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