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Identity & Inverse Matrices Identity What does “identity” mean to you? What is the multiplicative identity for the real numbers? In other words, 5 * __= 5? The identity for multiplication is 1 because anything multiplied by 1 will be itself. Inverses What does “inverse” mean to you? What is the inverse of multiplication? What do we multiply by to get the identity? In other words, 5 * ___=1? a * -1 a = 1 Any number multiplied by its inverse will be the identity. Identity Matrix The multiplicative identity for matrices is a square matrix with ones on the main diagonal and zeros everywhere else. 1 0 I 0 1 1 0 0 I 0 1 0 0 0 1 Identity Matrix Just like 5*1 = 5… AI= A 2 1 8 IA= A 3 1 0 2 17 0 1 1 8 3 17 Or 1 0 2 3 2 3 0 1 1 17 1 17 8 8 Identity Matrix Any matrix multiplied by its inverse will be the identity matrix. A * -1 A -1 A = I *A = I 2x2 Identity Matrix 1 0 I 0 1 3x3 Identity Matrix 1 0 0 I 0 1 0 0 0 1 Ex. 1 Determine whether A and B are inverses. 2 1 2 3 A 2 B 1 3 6 3 YES Ex. 2 Determine whether A and B are inverses. 4 3 5 3 B A 7 5 7 4 NO The Inverse of a 2x2 Matrix a b A c d -1 A = 1 A If ad-cd=0, then the matrix has no inverse!!!! d b c a 1 d b ad bc c a As long as ad-cb =0 Ex. 3 Find A-1, if it exists. 2 3 A 5 7 -1 A = 1 7 3 14 15 5 2 7 3 A 5 2 1 Ex. 4 Find A-1, if it exists. 2 1 A 4 0 -1 A = 1 0 0 1 1 4 1 4 4 2 1 2 Ex. 5 Find A-1, if it exists. 3 4 A 2 6 1 0 Does not exist, because it’s not square. Now let’s learn how to use our calculator!!! Find the inverse! 2 3 A 5 7 2 1 A 4 0 Yes, now you can add, subtract, multiply, and find the determinant in you calculator!! Solving Systems using Matrices and Inverses Solving Matrix Equations Suppose ax = b How do you solve for x? We cannot divide matrices, but we can multiply by the inverse. A-1 AX =A-1 B IX = A-1B X = A-1B Solving a Matrix Equation Solve the matrix equation AX=B for the 2x2 matrix X 4 1 8 5 X 3 1 6 3 X = A-1B 2 2 X 0 3 Ex. Solve 3 4 3 8 X 5 7 2 2 29 48 X 21 34 Solving Systems Using Inverse Matrices 5x 2 y 3 4x 2 y 4 Setting Up the Matrices • Matrix A will be the coefficients of the system • Matrix X will be the variables • Matrix B will be constants (what the system of equations are equal to) Matrix Equation A linear system can be written as a matrix equation AX=B 5 4 x 8 1 2 y 6 Constant Coefficient matrix Variable matrix matrix 5x 4 y 8 1x 2 y 6 Example 1 5x 4 y 8 1x 2 y 6 5 4 x 8 1 2 y 6 Example 2: Use matrices to solve the linear system 5x 2 y 3 4x 2 y 4 5 2 x 4 2 y 3 4 Type in [A]-1 [B] Find the inverse 1 1 3 x 5 y 2 4 2 (-1, 4) Example 3: Use matrices to solve the linear system 4x 2 y 8 x 2 y 12 4 2 x 8 1 2 y 12 Type in [A]-1 [B] Find the inverse x .2 .2 8 y .1 .4 12 (4, 4) Example 4: Use matrices to solve the linear system x y 2z 3 2 x y 3z 4 4 x 3 y z 18 2 x 3 1 1 2 1 3 y 4 4 3 1 z 18 Type in [A]-1 [B] (-2, 3, 1) Example 5: Use matrices to solve the linear system 2x z 2 5x y z 5 x 2 y 2z 0 2 0 1 x 2 5 1 1 y 5 1 2 2 z 0 Type in [A]-1 [B] (2, 3, -2) Let’s apply this… You have $18 to spend for lunch during a 5 day school week. It costs you $1.50 to make lunch at home and $5 to buy lunch. How many times each week do you make a lunch at home? x y 5 1.5 x 5 y 18 (2, 3) 1 1 x 5 1.5 5 y 18 Type in [A]-1 [B] You make lunch at home 2 times a week. A word problem…!! • A small corporation borrowed $1,500,000 to expand its product line. Some of the money was borrowed at 8%, some at 9% and some at 12%. How much was borrowed at each rate if the annual interest was $133,000 and the amount borrowed at 8% was 4 times the amount borrowed at 12%? $800,000 at 8% $500,000 at 9% $200,000 at 12% Homework