Identity and Inverse Matrices

Transcription

1 Identity and Inverse Matrices Booklet Four Copyright 207 Robert E. Mason IV. All Rights Reserved.

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3 Prepared by Dr. Robert E. Mason IV Mathematics Consultant 3

4 4 Identity and Inverse Matrices Objectives After studying this booklet, you should be able to: Write the identity matrix for any square matrix, and Find the inverse of a 2 x 2 matrix. Application: Crypotology You re sitting in math class reading a note that has been passe to you by your friend. Suddenly, the silence is broken by the teacher s voice saying, If that note is so interesting. Why don t you read it to the whole class? As you slowly walk to the front of the room you think, if I had written this note in code, I wouldn t be so embarrassed to read it! Cryptology deals with coding messages so that only people with the key can decipher them. In ancient Greece, Spartans wound a belt in spiral around a stick and wrote messages along the length of the stick. When they unwound the belt, only those people who had a stick exactly the same size as the first could read the message. Since then, cryptology has been important in military communications, particularly in time of war. Today, programmers use cryptology to protect secret data stored on computers. An important advancement in cryptology occurred in the 930s when American mathematician Lester Hill use matrices to encode messages. Here s a simplified version of how it works. Step Suppose the first word in a message is MEET. Assign each letter a number base on its position in the alphabet (A =, B =2,..., Y = 25, Z = 26). Thus, M = 3, E = 5, and T = 20.

5 5 Write the numbers in a matrix. M E E T = Step 2 Multiply the matrix by a coding matrix. Let s use! 0 $. Step = Assign a letter to each number of the matrix based on Step = E R T Y Therefore, MEET would encode as ETRY. Notice that each E in the encoded message is assigned to a different letter in the coded message. When the person received the message, he or she needs to decipher it by undoing the multiplication to get back to the original matrix. Mathematically, this means finding the inverse of the coding matrix. You will use the code in Example 4 Recall from your work with real numbers that the inverse and identity of a number are related. In real number. is the identity of multiplication because

6 6 a x = x a = a. Similary, the inverse of a matrix is related to the identity matrix. The identity matrix is a square matrix that, when multiplied by another matrix, equals the same matrix. With 2 x 2 matrices,! 0 $ is the identity matrix because!a b 0 c d $! 0 0 $ =! a b 0 $ and because! $!a b c d 0 c d $ =!a b $. The identity matrix is symbolized c d by I. Since one of the properties of the identity matrix is that it is commutative, only a square matrix can have an identity. In an identity matrix, the principal diagonal goes from upper left to lower right and consists only of ones. Identity Matrix for Multiplication The identity matrix for multiplication, I, is a square matrix with for every element of the principal diagonal and 0 in all other positions. For ay square matrix A of the same order as I, A I = I A = A. Example If N = , find I so that N x I = N. 8 2 The dimensions of N are 3 x 3. So, I must also be 3 x 3. The principal diagonal contains only s. Complete the matrix with 0s The 3 x 3 identity matrix is

7 = Therefore, N x I = N. Another property of real numbers is that every real number, except 0, has a multiplicative inverse. That is, 4 is the multiplicative inverse of a because 5 a 4 = 4 a =. Likewise, if matrix A has an inverse 5 5 A-, then A x A - = A - x A = I. The following example shows how the inverse of a 2 x 2 matrix can be found Example 2 If M =, find M -. Check your result w y x z = 0 0 7w + 4y 2w + 3y 7x + 4z 2x + 3z = 0 0 Multiply. When two matrices are equal, their corresponding elements are equal. So the following equations can be generated from the two equal matrices.. 7w + 4y = 2. 7x + 4z = w + 3y = x + 3z = Use equation () and (3) to find values of w and y. First solve for w. Then substitute the w value into one of the equations to find y.

8 8 7w + 4y = 2w +2y = 3 2w + 3y = 0 8w +2y = 0 3w = 3 w = 3 3 7w + 4y = y = 4y = y = Use equation (2) and (4) to find values for x and z. First solve for z. Then substitute the z value into one of the equations to find x. 7x + 4z = 0 4x + 8z = 0 2x + 3z = ( ) 4x + 2z = 7 M = -3z = z = Therefore.. 7x + 4z = 0 7x = 0 7x = x = Check: = or 0 0 The same method used in Example 2 can be used to develop the general form of the inverse of a 2 x 2 matrix

9 9 The inverse a c b d is d ad bc c ad bc b ad bc a ad bc or ad bc d c b a. Notice the ad - bc is the value of the determinant of the matrix. Remember that ad bc is not defined when ad - bc = 0. Therefore, if the value of the determinant of a matrix is 0, the matrix cannot have an inverse. Inverse of a 2 x 2 Matrix Any matrix M = a c b d a c 0. Then b d ad bc will have an inverse M - if and only if d c b a. Example 3 If Q = 2 3, find Q -. Check your result. Find the value of the determinant. 2 3 = 6 ( ) or 5 Since the determinant does not equal 0, Q - exists. Q = ad bc = d c b a

10 0 Check: = = 0 0 In the application at the beginning of the booklet, the coding matrix! 0 $ was used to encode the word MEET as ETRY. In the follow Example, you will decode part of the message by finding the inverse of the coding matrix. Example 4 Suppose a person receives the message ETRYATNYEMYULAMMIXCQ that has been encoded using the matrix C=! 0 $. You already know that first four letters, ETRY, correspond to MEET. Decode the next four letters in the message. First, write the letters in a 2 x 2 matrix and assign each letter a number based on its position in the alphabet. A N T Y = Now, find the inverse of the coding matrix C=! 0 $. The determinant is 0 - or -. C = 0 or 0 Finally, multiply the inverse matrix by the first matrix and assign letters to the elements in the product.

11 = or M A E T Therefore, the next four letters of the message are MEAT. The first eight letters of the message are MEETMEAT. You will decode the rest of the message in the exercises 6 and 27. Check for Understanding Study the lesson. Then complete the following in your toolbox-book.. Explain how the multiplicative inverse and identity for real numbers are similar to the matrix inverse and identity. 2. Write the 4 x 4 identity matrix. 3. Choose the inverse of! $ a b. 0 0 c d Create a square matrix that does not have an inverse. 5. You Decide Miyoki says that the matrix 4 does not have a multiplicative identity. Hector says the identity is 0 0 Guided Practice 4 = Who is correct? Explain your reasoning. because

12 6. Cryptology Decode the next four letters, EMYU, of the message in Example 4. 2

13 3 Excercises Find the inverse of each matrix, if it exists. If it does not exist, explain why not Determine whether each statement is true or false = I 7. = I I = = I 20. The inverse of is

14 4 2. All square matrices have multiplicative identities. 22. Only square matrices have multiplicative inverses. 23. Some square matrices do not have multiplicative inverses. 24. Some square matrices do not have multiplicative identities. 25. When Beth used a graphing calculator to find the inverse of there was an ERROR statement. Explain why Geometry Recall that the matrix 0 will rotate a figure on a coordinate plane 90-degrees counterclockwise about the origin. a. Find the inverse of this rotation matrix. b. Make a conjecture about what movement the inverse describes on a coordinate plane. c. Test your conjecture on the triangle below. Make a drawing to verify your conjecture Cryptology Refer to Example 4 and Exercise 6. a. Decode the last eight letters, LAMMIXCQ, of the coded message. (Hint: Negative integers and zero are assigned letters as follows, 0 =Z, - = Y, -2 = X, -3 = W, and so on.) b. Write the entire decoded message from Example 4. c. Write the message and code it using you own coding matrix. (Hint: Use a coding matrix whose determinant is or -.) Trade messages with a partner and decode the messages.

15 5 4 a 7 3a = Solve (Hint: read Example 2) 29. Solve the system of equation by using substitution. 3x 2y = -3 3x + y = Solve for the variables - 3x y = 6x 4y 7.

16 6 Test Your Understanding (Self-test) Identity and Inverse Matrices Determine whether or not the given matrices are inverses for each other and and Find the inverse, if it exists, for each matrix. Write all equations that you used to determine if the matrix has an inverse, and the equation for finding the inverse matrix (do this one time). Then replace the variables with the specific numbers in each problem and solve Solve each system of equations using the matrix inverse of the coefficient matrix. Must show detailed work for each problem x + y = 2x y = 2x y = 8 3x + y = 2

17 7 Matrices Inverses KEY Determine whether or not the given matrices are inverses for each other and 0 0 NO and NO Find the inverse, if it exists, for each matrix thesolutionis Solve each system of using the inverse of the coefficient matrix. 5. x + y = 2x y = {(2, 3)} 6. 2x y = 8 3x + y = 2 {(-2, 4)}

18 8 Challenge Matrices Problems Solve each matrix equation. X is not a scalar. X is a matrix X = X = = X = Find the coordinates of the vertices of the image of triangle ABC with A(4, 3), B(2,), and C(, 5) after it is rotated 90 degrees counterclockwise about the origin. 5. Triangle ABC with vertices A(, 4), B(2, -5), and C(-6, -6) is translated 3 units right and unit down.. Write the translation matrix. 2. Find the coordinates of triangle A B C. 3. Graph the preimage and the image. 6. The vertices of Δ ABC are A(-2,), B(,2), and C(2,-3). The triangle is dilated so that its perimeter is 2.5 times the original perimeter.. Write the coordinates of Δ ABC in a vertex matrix. 2. Find the coordinates of Δ A B C. 3. Graph Δ ABC and Δ A B C.

19 9 Challenge Matrices Problems -Key Solve in each matrix equation Find the coordinates of the vertices of the image of triangle ABC with A(4, 3), B(2,), and C(, 5) after it is rotated 90 degrees counterclockwise about the origin. The coordinates of the vertices of A B C are A (-3,4), B (-,2), and C (-5, ).

20 20 5. Triangle ABC with vertices A(, 4), B(2, -5), and C(-6, -6) is translated 3 units right and unit down a. Write the translation matrix. b. Find the coordinates of triangle. A B C. A (4,3), B (5,-6), C (-3, -7) c. Graph the preimage and the image.

21 2 6. The vertices of Δ ABC are A(-2,), B(,2), and C(2,-3). The triangle is dilated so that its perimeter is 2.5 times the original perimeter.. Write the coordinates of Δ ABC in a vertex matrix Find the coordinates of Δ A B C. A (-5, 2.5), B (2.5, 5), and C (5, -7.5) 3. Graph Δ ABC and Δ A B C.