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4.7 Identity and Inverse Matrices What is an identity? In math the identity is the number you multiply by to have equivalent numbers. For multiplication it is one. 5 * 1 = 5, of course. How is this useful? 3 If you need to have a denominator of 40 in the fraction 8 3 5 15 8 5 40 What is an inverse? The inverse is a number when multiplied by another number equals one. 1 4 1 4 4 1 4 1 4 4 4 9 36 1 9 4 36 The identity matrix is a square matrix with one down the diagonals In a 2 X 2 1 0 0 1 In a 3 X 3 1 0 0 0 1 0 0 0 1 Find the matrices that when multiply together are the identity matrix. How do we find the inverse? ? ? 2 8 1 0 ? ? 1 3 0 1 Find the inverse Find the determinant of 2 8 1 3 6 – ( - 8) = 14 We will flip the diagonal of top left and bottom right, then change the signs the bottom left to top right diagonal. 2 8 3 8 1 3 1 2 Putting the part together Take the determinant and put it under one and multiply it by the moved matrix. 1 3 8 14 1 2 Lets see if it works 1 3 8 2 8 14 1 2 1 3 1 6 8 24 24 1 14 0 14 2 2 8 6 14 0 14 1 0 0 1 Can there be matrices without inverses? Yes, when the determinate equals zero. 4 6 2 3 (12 (12)) 0 Since a fraction can not have zero for a denominator. There would be no inverse. Homework Page 199 # 10 - 30