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What is AI? 2 1 2 0 1 5 A 2 2 3 1 0 0 I 3 20 1 12 0 What is IA? 0 1 5 A 0 0 1 2 2 3 1 2 2 A 0 1 5 2 2 3 Multiplying a matrix by the identity gives the matrix back again. 1 0 0 I 3 0 1 0 0 0 1 an n n matrix with ones on the main diagonal and zeros elsewhere Can we find a matrix to multiply the first matrix by to get the identity? 1 3 1 1 2 1 ? 3 4 22 0 2 1 1 3 1 1 2 3 4 2 0 2 2 0 1 0 1 Let A be an n n matrix. If there exists a matrix B such that AB = BA = I then we call this matrix the inverse of A and denote it A-1. a b c d Finding the Inverse of a 2x2 matrix Step-1 First find what is called the Determinant This is calculated as ad-bc Step-2 Then swap the elements in the leading diagonal d b c a d b c a Step-3 Then negate the other elements Step-4 Then multiply the Matrix by 1/determinant 1 d b ad cb c a inverse matrix 4 Example Find Inverse of A Step 1 – Calc Determinant 4 8 Determinant (ad-cb) = 4x3-8x1 = 4 A 1 3 Step 2 – Swap Elements on leading diagonal step2 3 8 1 4 Step 3 – negate the other elements step3 3 8 4 1 Step 4 – multiply by 1/determinant step4 1 3 8 4 4 1 check 0.75 2 1 A 0 . 25 1 AA inverse matrix 1 4 8 0.75 2 1 3 0.25 1 3 2 0.75 0.75 8 8 1 0 0 1 23 5 Find the inverses and check them 2 6 A 1 5 1 5 6 1.25 1.5 A 4 1 2 0.25 0.5 5 20 B 1 2 1 2 20 0.2 2 B 10 1 5 0.1 0.5 2 2 C 0 1 1 1 1 1 2 0.5 1 C 2 0 1 2 0 1 inverse matrix 6 We can use A-1 to solve a system of equations x 3y 1 2x 5 y 3 To see how, we can re-write a system of equations as matrices. Ax b coefficient matrix variable matrix 1 3 2 5 x y constant matrix 1 3 Why will it help us solve equations? Because if we can express a system of equations in the form Ax b Then we can multiply both sides by the inverse matrix 1 1 A Ax A b And we can then know the values of X because 1 A A I x A b 1 inverse matrix 8 Solving systems of equations We can use our 2x2 matrices to express these systems of equations x y x 3y 240 0 Becomes in matrix form 1 1 x1 240 1 3 x 2 0 constants from the left hand side UNKNOWNS X ~ x1 Y ~ x2 inverse matrix constants from the right hand side 9 Your calculator can compute inverses and determinants of matrices. For the inverse: Step 1: enter the matrix into the calculator Step 2: Pull up matrix entered onto the screen Step 3: hit the x^-1 button and enter (this will give you the inverse matrix). Now using the previous problem we will allow the calculator to do the work. Enter this as matrix A in the calc 1 1 x1 240 1 3 x 2 0 Enter this as Matrix B in the calc Your Turn solve the following 3x +4y = 5 5x = 7-6y x+7y = 1.24 3y -x = 0.76 8x = 3y -1 x+y =-7 3 4 x 5 5 6 y 7 1 7 x 1.24 1 3 y 0.76 8 3 x 1 1 1 y 7 inverse matrix Answer x = -1 y = 2 Answer x = -0.16 y = 0.2 Answer x = -2 y = -5 12