1. Investigating Identity and Inverse Matrices QUESTION: What are some properties of identity and inverse matrices? 1 Let A =, B =, and C=. Consider the 2  2 identity matrix I =. Find AI, BI, and CI. What do you notice? 13251325 –40 –76 0.10.8 0.60.3 10011001 2Find IA, IB, and IC using the matrices from Step 1. Is multiplication by the identity matrix commutative? 3 Let D =. The inverse of D is E =. Find DE and ED. 75437543 3–5 –4 7 4Use matrix multiplication to decide which of the following is the inverse of the matrix A in Step 1 :,, or. 5–3 –2 1 –5 3 2–1 –1 2 3–5

2. Investigating Identity and Inverse Matrices DRAWING CONCLUSIONS For any 2  2 matrix A, what is true of the products AI and IA where I is the 2  2 identity matrix? Justify your answer mathematically. (Hint: Let A =, and compute AI and IA.) abcdabcd How is the relationship between I = and other 2  2 matrices similar to the relationship between 1 and other real numbers? 10011001 What do you think is the identity matrix for the set of 3  3 matrices? Check your answer by multiplying your proposed identity matrix by several 3  3 matrices.

3. Investigating Identity and Inverse Matrices DRAWING CONCLUSIONS What is the relationship between a matrix, its inverse, and the identity matrix? How is this relationship like the one that exists between a nonzero real number, its reciprocal, and 1 ? Does every nonzero matrix have an inverse? Explain. (Hint: Consider a 2  2 matrix whose first row contains all nonzero entries and whose second row contains all zero entries.) 27142714 Find the inverse of F = by finding values of a, b, c, and d such that 27142714 abcdabcd 10011001 =.

4. Two n  n matrices are inverses of each other if their product (in both orders) is the n  n identity matrix. For example, matrices A and B below are inverses of each other. The number 1 is a multiplicative identity for real numbers because 1 a = a and a 1 = a. For matrices, the n  n identity matrix is the matrix that has 1 ’s on the main diagonal and 0 ’s elsewhere. 2  2 Identity Matrix 3  3 Identity Matrix I =I = I = 10011001 100010001100010001 If A is any n  n matrix and I is the n  n identity matrix, then IA = A and AI = A. AB 21532153 = 10011001 =I BA 3–1 –5 2 = 10011001 =I The symbol used for the inverse of A is A –1. 3–1 –5 2 = 21532153 = Using Inverse Matrices

5. The number 1 is a multiplicative identity for real numbers because 1 a = a and a 1 = a. For matrices, the n  n identity matrix is the matrix that has 1 ’s on the main diagonal and 0 ’s elsewhere. THE INVERSE OF A 2  2 MATRIX The inverse of the matrix A = is abcdabcd A –1 = = provided a d – c b  0. d–b –c a d–b –c a 1 | A| 1 a d – c b Using Inverse Matrices

6. Finding the Inverse of a 2  2 Matrix Find the inverse of A =. 31423142 SOLUTION A –1 C HECK You can check the inverse by showing that AA –1 = I = A –1 A. 31423142 1 –2 1212 3232 1 –2 1212 3232 31423142 2–1 –4 3 1 6 – 4 = 2–1 –4 3 1212 = 1 –2 1212 3232 = = 10011001 and 10011001 =

7. Solving a Matrix Equation Solve the matrix equation AX = B for the 2  2 matrix X. AB 4–1 –3 1 8–5 –6 3 X = SOLUTION Begin by finding the inverse of A. A –1 =1 113411341 11341134 1 4 – 3 =

8. Solving a Matrix Equation To solve the equation for X, multiply both sides of the equation by A –1 on the left.1 113411341 11341134 4–1 –3 1 X= 8–5 –6 3 10011001 X= 2–2 0–3 X= 2–2 0–3 X = A –1 B IX = A –1 B A –1 AX = A –1 B C HECK You can check the solution by multiplying A and X to see if you get B. Solve the matrix equation AX = B for the 2  2 matrix X. AB 4–1 –3 1 8–5 –6 3 X =

9. If det A = 0, then A does not have an inverse. If det A  0, then A has an inverse. The inverse of a 3  3 matrix is difficult to compute by hand. A calculator that will compute inverse matrices is useful in this case. Using Inverse Matrices Some matrices do not have an inverse. You can tell whether a matrix has an inverse by evaluating its determinant.

10. Using Inverse Matrices in Real Life A cryptogram is a message written according to a secret code. (The Greek word kruptos means hidden and the Greek word gramma means letter.) The following technique uses matrices to encode and decode messages. First assign a number to each letter in the alphabet with 0 assigned to a blank space. _ = 0 E = 5 J = 10 O = 15 T = 20 Y = 25 A = 1 F = 6 K = 11 P = 16 U = 21 Z = 26 B = 2 G = 7 L = 12 Q = 17 V = 22 C = 3 H = 8 M = 13 R = 18 W = 23 D = 4 I = 9 N = 14 S = 19 X = 24 Then convert the message to numbers partitioned into 1  2 uncoded row matrices. To encode a message, choose a 2  2 matrix A that has an inverse and multiply the uncoded row matrices by A on the right to obtain coded row matrices.

11. Converting a Message Use the list below to convert the message GET HELP to row matrices. SOLUTION GET_HELP [7 5][20 0][8 5][12 16] _ = 0 E = 5 J = 10 O = 15 T = 20 Y = 25 A = 1 F = 6 K = 11 P = 16 U = 21 Z = 26 B = 2 G = 7 L = 12 Q = 17 V = 22 C = 3 H = 8 M = 13 R = 18 W = 23 D = 4 I = 9 N = 14 S = 19 X = 24

12. SOLUTION Encoding a Message CRYPTOGRAPHY Use A = to encode the message GET HELP. 2 3 –1–2 The coded row matrices are obtained by multiplying each of the uncoded row matrices from the previous example by the matrix A on the right. UNCODED ROW MATRIX ENCODING MATRIX A CODED ROW MATRIX [7 5] [20 0] [8 5] [12 16] 2 3 –1–2 2 3 –1–2 2 3 –1–2 2 3 –1–2 = [9 11] = [40 60] = [11 14] = [8 4] The coded message is 9, 11, 40, 60, 11, 14, 8, 4.

13. SOLUTION CRYPTOGRAPHY Use the inverse of A = to decode this message: 3–1 –2 1 Decoding a Message First find A –1 : =1 11231123 To decode the message, partition it into groups of two numbers to form coded row matrices. Then multiply each coded row matrix by A –1 on the right to obtain the uncoded row matrices. 1 11231123 1 3 – 2 A –1 = DECODING USING MATRICES For an authorized decoder who knows the matrix A, decoding is simple. The receiver only needs to multiply the coded row matrices by A –1 on the right to retrieve the uncoded row matrices.

14. Decoding a Message CODED ROW MATRIXDECODING MATRIX A –1 [– 4 3] [–23 12] [–26 13] [15 –5] 11 23 11 23 11 23 11 23 = [2 5] = [1 13] = [0 13] = [5 0] UNCODED ROW MATRIX CRYPTOGRAPHY Use the inverse of A = to decode this message: 3–1 –2 1

15. Decoding a Message [31 –5] [–38 19] [–21 12] [20 0] 11 23 11 23 11 23 11 23 = [21 16] = [0 19] = [3 15] = [20 20] CRYPTOGRAPHY Use the inverse of A = to decode this message: 3–1 –2 1 CODED ROW MATRIXDECODING MATRIX A –1 UNCODED ROW MATRIX

16. Decoding a Message [75 –25] 11 23 = [25 0] The uncoded row matrices are as follows. [2 5][1 13][0 13][5 0][21 16][0 19][3 15][20 20][25 0] CRYPTOGRAPHY Use the inverse of A = to decode this message: 3–1 –2 1 CODED ROW MATRIXDECODING MATRIX A –1 UNCODED ROW MATRIX

17. Decoding a Message CRYPTOGRAPHY Use the inverse of A = to decode this message: 3–1 –2 1 [2 5][1 13][0 13][5 0][21 16][0 19][3 15][20 20][25 0] _ = 0 E = 5 J = 10 O = 15 T = 20 Y = 25 A = 1 F = 6 K = 11 P = 16 U = 21 Z = 26 B = 2 G = 7 L = 12 Q = 17 V = 22 C = 3 H = 8 M = 13 R = 18 W = 23 D = 4 I = 9 N = 14 S = 19 X = 24 You can read the message as follows: BEAM_ME_UP_SCOTTY_