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Transcript
Section 9.4 Multiplicative Inverses of Matices and Matrix Equations The Multiplicative Identity Matrix 1 0 The Multiplicative Identity matrix is I= for 2 2 matrices. 0 1 That means that AI=A and IA=A The Multiplicative Inverse of a Matrix If a square matrix has a multiplicative inverse, it is said to be invertible or nonsingular. If a square matrix has a multiplicative inverse, the inverse is unique. If a square matrix has no multiplicative inverse, it is called singular. Example 2 1 Find the multiplicative inverse of A= 7 4 A Quick Method for Finding the Multiplicative Inverse of a 2 x 2 Matrix Example Find the multiplicative inverse of A using the Quick Method to find the inverse. 3 2 A= 1 4 Example Find the multiplicative inverse of A using the Quick Method. Check your work using your 2 3 calculator. A= 1 5 Example Show that A does not have an inverse. First by calculations, then use your calculator and see what you get for an answer. 3 2 A= 6 4 Finding Multiplicative Inverses of n x n Matrices with n Greater Than 2 If ad-bc=0 then the matrix has no multiplicative inverse. Example Find the inverse function without a calculator for 2 1 1 1 A= . Show that A A I and A A=I 2 . 2 1 3 Find the multiplicative inverse of A by row calculations, then check your work using the calculator. 1 1 0 A= 1 3 4 . 0 4 3 Example Find the multiplicative inverse matrix of A using row calculations. Then check your answer using your calculator. 1 2 2 A= 0 1 1 . 2 1 0 Solving Systems of Equations Using Multiplicative Inverses of Matrices Solve the system using A 1 , the inverse of the coefficient matrix. x+z=3 x-y=-2 x-y+2z=2 Example Solve the system by using A 1 , the inverse of the coefficient matrix. x+ y- z =2 2y+ z=3 x 2 y 1 Applications of Matrix Inverses to Coding 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 ATTITUDE=1,20,20,9,20,21,4,5 The numerical equivalent of the word MATH is 13,1,20,8. The numerical equivalent of the message is then converted into a matrix. Finally, an invertible matrix can be used to convert the message into code. The multiplicative inverse of this matrix can be used to decode the message. Encoding the Word MATH Decoding a Word Example 4 1 For the word CASH which is 3,1,19,8, use the coding matrix A= 3 1 1 1 to encode the word. Then use the matrix A 1 to decode the given 3 4 word. The problem has already been started for you. 4 1 3 19 3 1 1 8 1 3 Find the multiplicative inverse of A= . 2 2 2 3 (a) 2 1 1 0 (b) 0 1 1 3 (c) 4 8 1 1 4 8 1 3 (d) 2 4 1 1 2 4 1 3 1 Find the multiplicative inverse of A= . 2 2 1 (a) 2 3 1 2 1 1 1 0 2 (b) 0 1 1 1 4 1 (c) 1 2 0 (d) No inverse exists