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Bellwork 1) Multiply. 3 3 0 3 2 1 2 1 0 1 2) Multiply. 2 9 6 2 3 4 3 3 4 2 2 1 3) Find the determinant. 12 2 5 8 4) Find the determinant 15 4 10 10 0 6 8 2 14 Answers 9 3 7 2 2 1 13 3.)106 4.) 12 -1 - 4 3.8A Use Inverse Matrices to solve linear systems Algebra II 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. 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 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 Identity Matrix Just like 5*1 = 5… AI= A 2 1 8 IA= A 3 1 0 2 1 17 0 1 8 3 17 Or 1 0 2 3 2 3 0 1 1 17 1 17 8 8 Determine whether A and B are inverses. 1 3 1 1 2 A B 3 4 2 2 2 Find out if AB = I If so, then yes they’re inverses. If not, then no they’re not inverses. Use your calculator. YES The Inverse of a 2x2 Matrix a b A c d -1 A = 1 A d b 1 d b c a ad bc c a As long as ad-cb =0 If ad-cb=0, then the matrix has no inverse!!!! Ex. 1 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. 2 Find A-1, if it exists. 3 4 A 2 6 1 0 Does not exist, because it’s not square. 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 Ex. 3 Solving a Matrix Equation Solve the matrix equation AX=B for the 2x2 matrix X 4 1 8 5 X 3 1 6 3 4 1 8 5 3 1 X 6 3 Step 1: find A-1 A-1= 1 1 1 1 1 1 4 3 3 4 3 4 Step 2: Multiply both sides of the equation By A-1 on the left 1 1 4 1 1 1 8 5 3 4 3 1 X 3 4 6 3 1 1 4 1 1 1 8 5 3 4 3 1 X 3 4 6 3 53 4 3 1 1 86 12 12 3 4 X 24 24 15 12 1 0 2 2 0 1 X 0 3 You can check the Solution by mult AX to see if you get B 2 2 X 0 3 Assignment Inverses What does “inverse” mean to you? What is the inverse of multiplication? Determine whether A and B are inverses. 5 3 A 7 4 4 3 B 7 5 NO