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Name: ______________________________________ Period ______ Sec6-3 Matrix Equations In this lesson you will: (1) (2) (3) (4) find an identity matrix find the inverse of a matrix write matrix equations use inverse matrices to solve matrix equations Start by reviewing what you’ve learned about matrices: Scalar Multiplication and Matrix Multiplication (a) 8 10 5 5 (b) 8 9 3 2 3 9 0 6 1 2 0 (c) 5 2 6 1 1 4 7 (d) Adding, Subtracting, 10 5 5 3 2 3 6 1 2 1 4 1 2 3 4 5 6 2 5 3 6 Find the missing values. SHOW WORK 4 a 1 10 b 5 2 20 Identity Matrix: In earlier math courses, you learned that the number 1 is the multiplicative identity. This means that when you multiply any real number by 1, the number does not change. Similarly, when you multiply a matrix by an identity matrix, the matrix does not change. 2 1 a b Find the identity matrix for matrix means you need to find the matrix c d so that 4 3 2 1 a b 2 1 The product equals 4 3 c d 4 3 Find the values for a, b, c, d by using matrix multiplication. In general, an identity matrix is a square matrix with 1’s along the main diagonal from top left to bottom right, and 0’s for all the other entries. We use and “I” to name the identity matrix. 1 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 I= or I= or I= and so on … 0 0 1 0 0 1 0 0 1 0 0 0 1 The Inverse Matrix: In earlier math course you also learned that every nonzero real number has a multiplicative inverse, the number you multiply it by to get the multiplicative identity, 1. For example: The multiplicative inverse of 4 is ¼ because (4)(1/4) = 1 Similarly, SOME (but not all) SQUARE matrices have an inverse matrix. We write the inverse of a matrix A as A-1. when you multiply a matrix on either side by its inverse matrix you get the identity matrix. This means that: A A-1=I and also A-1A=I 2 3 4 3 (a) Are these two matrices inverses of each other? 1 4 1 2 7 5 2 5 (b) Are these two matrices inverses of each other? 3 2 3 7 2 1 (c) Find the inverse of matrix A= 4 3 2 1 a b 1 0 You will need to solve for a, b, c, d by using matrix multiplication: c d = 0 1 4 3 Confirm that you have the correct inverse matrix by using your calculator to multiply matrix A with it’s inverse. Another way to confirm that you found the correct inverse is to let the calculator find the 1 2 1 -1 inverse matrix: A = 4 3 Write a matrix equation to represent a system of equation: x 2y 1 3x 4 y 2 Coefficient Variable Constant Solve the matrix equation by using the Inverse Matrix: Steps: (1) Find the inverse of the coefficient matrix. (2) Multiply both sides of the equation by the Inverse Matrix (3) Simplify to solve for the variable matrix. Practice Problems 2 x 5 y 8 (1) Write a matrix equation and then solve by using the Inverse Matrix. 4x y 6 2x y 2z 1 (2) Write a matrix equation and then solve by using the Inverse Matrix. 6 x 2 y 4 z 3 4 x y 3z 5 (3) Find a, b, c such that the graph of (2, -5), (4, 21) and (-2, 15) y ax 2 bx c passes through the points: