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? –40 – Find 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 –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

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? 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.

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.) Find the inverse of F = by finding values of a, b, c, and d such that abcdabcd =.

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 = If A is any n  n matrix and I is the n  n identity matrix, then IA = A and AI = A. AB = =I BA 3–1 –5 2 = =I The symbol used for the inverse of A is A –1. 3–1 –5 2 = = Using Inverse Matrices

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

Finding the Inverse of a 2  2 Matrix Find the inverse of A = SOLUTION A –1 C HECK You can check the inverse by showing that AA –1 = I = A –1 A – – –1 – – 4 = 2–1 – = 1 – = = and =

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 = – 3 =

Solving a Matrix Equation To solve the equation for X, multiply both sides of the equation by A –1 on the left –1 –3 1 X= 8–5 – 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 =

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.

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.

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

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.

SOLUTION CRYPTOGRAPHY Use the inverse of A = to decode this message: 3–1 –2 1 Decoding a Message First find A –1 : = 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 – 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.

Decoding a Message CODED ROW MATRIXDECODING MATRIX A –1 [– 4 3] [–23 12] [–26 13] [15 –5] = [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

Decoding a Message [31 –5] [–38 19] [–21 12] [20 0] = [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

Decoding a Message [75 –25] = [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

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_