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

7.5 Determinants and Cramer’s Rule Subscript notation for the matrix A The row 1, column 1 element is a11; the row 2, column 3 element is a23; and, in general, the row i, column j element is aij.

7.5 Determinants of 2 × 2 Matrices Associated with every square matrix is a real number called the determinant of A. In this text, we use det A. The determinant of a 2 × 2 matrix A, is defined as

7.5 Determinants of 2 × 2 Matrices Example Find det A if Analytic Solution Graphing Calculator Solution

7.5 Determinant of a 3 × 3 Matrix The determinant of a 3 × 3 matrix A, is defined as

7.5 Determinant of a 3 × 3 Matrix A method for calculating 3 × 3 determinants is found by re-arranging and factoring this formula. Each of the quantities in parentheses represents the determinant of a 2 × 2 matrix that is part of the 3 × 3 matrix remaining when the row and column of the multiplier are eliminated.

7.5 The Minor of an Element The determinant of each 3 × 3 matrix is called a minor of the associated element. The symbol Mij represents the minor when the ith row and jth column are eliminated.

7.5 The Cofactor of an Element To find the determinant of a 3 × 3 or larger square matrix: Choose any row or column, Multiply the minor of each element in that row or column by a +1 or –1, depending on whether the sum of i + j is even or odd, Then, multiply each cofactor by its corresponding element in the matrix and find the sum of these products. This sum is the determinant of the matrix. Let Mij be the minor for element aij in an n × n matrix. The cofactor of aij, written Aij, is

7.5 Finding the Determinant Example Evaluate det , expanding by the second column. Solution First find the minors of each element in the second column.

7.5 Finding the Determinant Now, find the cofactor. The determinant is found by multiplying each cofactor by its corresponding element in the matrix and finding the sum of these products.

7.5 Cramer’s Rule for 2 × 2 Systems Note: Cramer’s rule does not apply if D = 0. For the system where, if possible,

7.5 Applying Cramer’s Rule to a System with Two Equations Example Use Cramer’s rule to solve the system. Analytic Solution By Cramer’s rule,

7.5 Applying Cramer’s Rule to a System with Two Equations The solution set is Graphing Calculator Solution Enter D, Dx, and Dy as matrices A, B, and C, respectively.

7.5 Cramer’s Rule for a System with Three Equations For the system where