Chapter 7: Matrices and Systems of Equations and Inequalities 7.2 Solution of Linear Systems in Three Variables 7.3 Solution of Linear Systems by Row Transformations 7.4 Matrix Properties and Operations 7.5 Determinants and Cramer’s Rule 7.6 Solution of Linear Systems by Matrix Inverses 7.7 Systems of Inequalities and Linear Programming 7.8 Partial Fractions

7.6 Solution of Linear Systems by Matrix Inverses If there is a multiplicative identity matrix I, such that for any matrix A, then A and I must be square matrices of the same dimensions. The 2 × 2 identity matrix, denoted I2, is This is easily verified by showing A I2 = A and I2 A = A, for any 2 × 2 matrix A.

7.6 Using the 3 × 3 Identity Matrix I3 Example Show that Graphing Calculator Solution Using

7.6 Multiplicative Inverses of Square Matrices Suppose If AB = I2 and BA = I2, then B is the inverse of A. Inverse of a 2 × 2 matrix

7.6 The Inverse of a 2 × 2 Matrix Solving the system yields We can show that AB = I2 and BA = I2.. Thus, we can conclude that B is the inverse of A, written A-1, provided that the det A  0.

7.6 The Inverse of a 2 × 2 Matrix If and det A  0, then or

7.6 Finding the Inverse of a 2 × 2 Matrix Example Find A-1 if it exists. Analytic Solution (a) (b) Here, A-1 does not exist.

7.6 Finding the Inverse of a 2 × 2 Matrix Graphing Calculator Solution (a) (b) The calculator returns a singular matrix error when directed to find the inverse of a matrix whose determinant is 0.

7.6 Solving Linear Systems Using Inverse Matrices Solve the system AX = B, where A is the coefficient matrix, X is the matrix of variables, and B is the matrix of the constants. Note: When multiplying by matrices on both sides, multiply in the same order on both sides. Multiply both sides by A-1. Associative property Multiplicative inverse property Identity property

7.6 Solving Linear Systems Using Inverse Matrices Example Solve the system using the inverse of the coefficient matrix. Analytic Solution

7.6 Solving Linear Systems Using Inverse Matrices The solution set {(x, y, z)} = {(4, 2, –3)}.

7.6 Solving Linear Systems Using Inverse Matrices Graphing Calculator Solution Enter the coefficient matrix A and the constant matrix B. Make sure the det A  0. The solution verifies the results achieved analytically.