Study Guide and Intervention. Solving Systems of Equations Using Inverse Matrices

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1 -8 Study Guide and Intervention Identity and Inverse Matrices The identity matrix for matrix multiplication is a square matrix with s for every element of the main diagonal and zeros elsewhere. Identity Matrix for Multiplication If A is an n n matrix and I is the identity matrix, then A I = A and I A = A. If an n n matrix A has an inverse A -, then A A - = A - A = I. Example Determine whether X = inverse matrices. 0 Find X Y. X Y = 0 = Find Y X. Y X = -5 = or or Since X Y = Y X = I, X and Y are inverse matrices. Exercises 4 6 and Y = -5 - are Determine whether the matrices in each pair are inverses of each other and 4-4 and -4 - and and and and 8 6 and - 5 and and and 4 and 5 and Chapter 50 Glencoe Algebra

2 -8 Study Guide and Intervention (continued) Matrix Equations A matrix equation for a system of equations consists of the product of the coefficient and variable matrices on the left and the constant matrix on the right of the equals sign. Example x - y = x + 5y = -8 Determine the coefficient, variable, and constant matrices. - 5 x y = -8 Find the inverse of the coefficient matrix. 5 5 (5) - (-) - = - Rewrite the equation in the form of X = A - B 5 x y = - -8 Solve. x y = - 8 Exercises Use a matrix equation to solve a system of equations. Use a matrix equation to solve each system of equations.. x + y = 8. 4x - y = 8 5x - y = - x + y =. x - y = x - 6y = 0 x + y = -0 x + y + 8= 0 Lesson x + y = 8 6. x - y = 4 x = -4y + 5 y = 80 - x Chapter 5 Glencoe Algebra

3 -8 Skills Practice Determine whether the matrices in each pair are inverses.. X = 0, Y = - 0. P =, Q = - -. M = - 0 0, N = A = - - 5, B = V = 0-0, W = X = - 4, Y = G = 4 -, H = D = , E = Find the inverse of each matrix, if it exists Use a matrix equation to solve each system of equations. 5. p - q = x - y = p + q = -6-4x - 5y =. m + n = a + b = -9 6m + 4n = -8 5a - b = 4 Chapter 5 Glencoe Algebra

4 -8 Practice Determine whether each pair of matrices are inverses.. M =, N = -. X = A = - -4, B = P = Determine whether each statement is true or false. 5. All square matrices have multiplicative inverses. 6. All square matrices have multiplicative identities. -, Y = 5 -, Q = 4 Find the inverse of each matrix, if it exists Lesson -8. GEOMETRY Use the figure at the right. a. Write the vertex matrix A for the rectangle. b. Use matrix multiplication to find BA if B = c. Graph the vertices of the transformed quadrilateral on the previous graph. Describe the transformation. d. Make a conjecture about what transformation B - describes on a coordinate plane. 4. CODES Use the alphabet table below and the inverse of coding matrix C = to decode this message: CODE A B C D 4 E 5 F 6 G H 8 I 9 J 0 K L M N 4 O 5 P 6 Q R 8 S 9 T 0 U V W X 4 Y 5 Z 6 0 y (, ) O (, ) (4, 4) (5, ) x Chapter 5 Glencoe Algebra

5 -8 Word Problem Practice. TEACHING Paula is explaining matrices to her father. She writes down the following system of equations. x + y = 4 x + y = 5. Next, Paula shows her father the matrices that correspond to this system of equations. What are the matrices?. AGES Hank, Laura, and Ned are ages h, l, and n, respectively. The sum of their ages is 5 years. Laura is one year younger than the sum of Hank and Ned s ages. Ned is three times as old as Hank. Use matrices to determine the age of each sibling.. TRANSPORTATION Paula wrote the following matrix equation to show the costs of two trips by water taxi to Logan Airport in Boston. She used x for the cost of round trips and y for the cost one one-way trips. x y = 6 54 Next, she found the inverse. - = Then she computed her answer. x y = = 4 0 When she checked her answer, the total cost of the trips came out as $ and $88. Where did she make a mistake? 4. SELF-INVERSES Phillip notices that any matrix with ones and negative ones on the diagonal and zeroes everywhere else has the property that it is its own inverse. Give an example of a by matrix that is its own inverse but has at least nonzero number off the diagonal. 5. MATRIX OPERATIONS Garth is studying determinants and inverses of matrices in math class. His teacher suggests that there are some matrices with unique properties, and challenges the class to find such matrices and describe the properties found. Garth is curious about the matrix G = a. What is the determinant of G? b. Does the inverse of G exist? Explain. c. Determine a matrix operation that could be used to transform G into its Additive Identity matrix. Chapter 54 Glencoe Algebra

6 -8 Enrichment Permutation Matrices A permutation matrix is a square matrix in which each row and each column has one entry that is. All the other entries are 0. Find the inverse of a permutation matrix by interchanging the rows and columns. For example, row is interchanged with column, row is interchanged with column P = P - = P is a 4 4 permutation matrix. P - is the inverse of P. Solve each problem.. There is just one permutation. Find the inverse of the matrix you wrote matrix that is not also an identity in Exercise. What do you notice? matrix. Write this matrix.. Show that the two matrices in Exercises and are inverses. Lesson Write the inverse of this matrix. 0 0 B = Use B - from problem 4. Verify that B and B - are inverses. 6. Permutation matrices can be used to write and decipher codes. To see how this is done, use the message matrix M and matrix B from problem 4. Find matrix C so that C equals the product MB. Use the rules below. 0 times a letter = 0 times a letter = the same letter 0 plus a letter = the same letter S H E M = S A W H I M. Now find the product CB -. What do you notice? Chapter 55 Glencoe Algebra