Avon High School Section: 9.4 ACE COLLEGE ALGEBRA II - NOTES Multiplicative Inverses of Matrices and Matrix Equations Mr. Record: Room ALC-129 Semester 2 - Day 17 The Multiplicative Identity Matrix 1 3 1 0 What happens if we multiply ? 4 6 0 1 1 1 0 0 0 1 0 Matrices like , 0 1 0 , 0 1 0 0 1 0 0 0 0 0 1 0 0 are all called identity matrices. 0 1 0 0 0 1 The Multiplicative Inverse of a Matrix Definition of the Multiplicative Inverse of a Square Matrix Let A be an m n matrix. If there exists an n n matrix and A1 (read: “A inverse”) such that AA1 I n and A1 A I n , then A1 is the multiplicative inverse of A. Example 1 The Multiplicative Inverse of a Matrix Show that B is the multiplicative inverse of A, where 2 1 1 1 A and B . 1 1 1 2 Example 2 Finding the Multiplicative Inverse of a Matrix Find the multiplicative inverse of 5 7 A 2 3 A Quick Method for Finding the Multiplicative Inverse of a 2 x 2 Matrix Multiplicative Inverse of a 2 x 2 Matrix a b 1 d b , then A1 . If A ad bc c a c d The matrix A is invertible if and only if ad bc 0 . If ad bc 0, then A does not have a multiplicative inverse. Example 3 Using the Quick Method to Find the Multiplicative Inverse of a Matrix Find the multiplicative inverse of 3 2 A 1 1 Finding Multiplicative Inverses of a n x n Matrices with n Greater Than 2 Example 4 Finding the Multiplicative Inverse of a 3 x 3 Matrix Find the multiplicative inverse of 1 0 2 A 1 2 3 1 1 0 Solving Systems of Equations Using Multiplicative Inverses of Matrices Solving a System Using A-1 If AX B has a unique solution, X A1B. To solve a linear system of equations, multiply A1 and B to find X. Example 5 Using the Inverse of a Matrix to Solve a System Solve the system by using A1 , the inverse of the coefficient matrix: x 2z 6 x 2 y 3 z 5 x y 6 Avon High School Section: 9.4 ACE COLLEGE ALGEBRA II - NOTES Multiplicative Inverses of Matrices and Matrix Equations Mr. Record: Room ALC-129 Semester 2 - Day 18 Applications of Matrix Inverses to Coding 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 Encod 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 Scalar codedMultiplication matrix. 4. Use the numbers, by columns, from the coded matrix in Step 3 to write the encoded message. Example 6 Encoding a Word 1 3 4 a. Encode the message MATH IS COOL using the coding matrix A 2 1 3 . Use 0 for blank. 4 2 1 b. Decode the message 138, 81, 102, 101, 67, 109, 162, 124, 173, 210, 150, 165 1 3 4 Given the coding matrix A 2 1 3 and a TI-Nspire calculator. 4 2 1