Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall

Chapter 5 More Work with Matrices

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 5.5 Matrix Equations

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Matrix multiplication can be used to represent a system of linear equations. The matrix equation is abbreviated as AX = B, where A is the coefficient matrix of the system, and X and B are matrices containing one column, called column matrices. The matrix B is called the constant matrix.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Solving a System Using A  1 If AX = B has a unique solution, then X = A  1 B. To solve a linear system of equations, multiply A  1 and B to find X.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 1 Solve the system by using A  1, the inverse of the coefficient matrix. Solution Write the linear system. The inverse was found in a previous section. continue

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Solve the system by using A  1, the inverse of the coefficient matrix. Write the linear system. Thus, x = 1/2, y =  1/2, and z = 1. The solution is

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall A cryptogram is a message written so that no one other than the intended recipient can understand it. To encode a message we begin by assigning a number to each letter in the alphabet: A = 1, B = 2, C = 3, …Z = 26, and a space = 0. The numerical equivalent of the word MATH is 13, 1, 20, 8.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Encoding a Word or Message 1. Express the word or message numerically. 2. List the numbers in step 1 by columns and form a square matrix. If you do not have enough numbers to form a square matrix, put zeros in any remaining spaces in the last column. 3. Select any square invertible matrix, called the coding matrix, the same size as the matrix in step 2. Multiply the coding matrix by the square matrix that expresses the message numerically. The resulting matrix is the coded matrix. 4. Use the numbers, by columns, from the coded matrix in step 3 to write the encoded message.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 2 Use matrices to encode the word MATH. Solution 1. Express the word numerically. MATH = 13, 1, 20, 8 2. List the numbers in step 1 by columns and form a square matrix. 3. Multiply the matrix in step 2 by a square invertible matrix of your choice. We will use the following as the coding matrix. continue

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Use matrices to encode the word MATH. 4. Use the numbers, by columns, from the coded matrix in step 3 to write the encode message. The encoded message is  29, 43,  64, 92.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall The inverse of coding a matrix can be used to decode a word or message that was encoded. Decoding a Word or Message That Was Encoded 1. Find the multiplicative inverse of the coding matrix. 2. Multiply the multiplicative inverse of the coding matrix and the coded matrix. 3. Express the numbers, by columns, from the matrix in step 2 as letters.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 3 Decode  29, 43,  64, 92 from Example 2. Solution 1. Find the inverse of the coding matrix. The coding matrix in Example 2 was We use the formula for the multiplicative inverse of a 2  2 matrix to find the multiplicative inverse of the matrix. continue

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Decode  29, 43,  64, 92 from Example Multiply the multiplicative inverse of the coding matrix and the coded matrix. 3. Express the numbers, by columns, from the matrix in step 2 as letters. 13, 20, 1, 8 = MATH