Determinants Every n  n matrix A is associated with a real number called the determinant of A, written  A . The determinant of a 2  2 matrix is defined as follows.

Determinant of a 2  2 Matrix If A =

Note Matrices are enclosed with square brackets, while determinants are denoted with vertical bars. A matrix is an array of numbers, but its determinant is a single number.

Determinants The arrows in the following diagram will remind you which products to find when evaluating a 2  2 determinant.

Example 1 EVALUATING A 2  2 DETERMINANT Let A = Find  A . Use the definition with Solution a 11 a 22 a 21 a 12

Determinant of a 3  3 Matrix If A =

Evaluating The terms on the right side of the equation in the definition of  A  can be rearranged to get Each quantity in parentheses represents the determinant of a 2  2 matrix that is the part of the matrix remaining when the row and column of the multiplier are eliminated, as shown in the next slide.

Evaluating

Cramer’s Rule for Two Equations in Two Variables Given the system if then the system has the unique solution where

Caution As indicated in the preceding box, Cramer’s rule does not apply if D = 0. When D = 0 the system is inconsistent or has infinitely many solutions. For this reason, evaluate D first.

Example 4 APPLYING CRAMER’S RULE TO A 2  2 SYSTEM Use Cramer’s rule to solve the system Solution By Cramer’s rule, and Find D first, since if D = 0, Cramer’s rule does not apply. If D ≠ 0, then find D x and D y.

Example 4 APPLYING CRAMER’S RULE TO A 2  2 SYSTEM By Cramer’s rule, The solution set is as can be verified by substituting in the given system.

General form of Cramer’s Rule Let an n  n system have linear equations of the form Define D as the determinant of the n  n matrix of all coefficients of the variables. Define D x1 as the determinant obtained from D by replacing the entries in column 1 of D with the constants of the system. Define D xi as the determinant obtained from D by replacing the entries in column i with the constants of the system. If D  0, the unique solution of the system is

Example 5 APPLYING CRAMER’S RULE TO A 3  3 SYSTEM Use Cramer’s rule to solve the system. Solution Rewrite each equation in the form ax + by + cz +  = k.

Example 5 APPLYING CRAMER’S RULE TO A 3  3 SYSTEM Verify that the required determinants are

Example 5 APPLYING CRAMER’S RULE TO A 3  3 SYSTEM Thus, and so the solution set is

Note When D = 0, the system is either inconsistent or has infinitely many solutions. Use the elimination method to tell which is the case. Verify that the system in Example 6 is inconsistent, so the solution set is ø.