College Algebra Chapter 6 Matrices and Determinants and Applications Section 6.5 Determinants and Cramer’s Rule Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

Concepts Evaluate the Determinant of a 2 × 2 Matrix Evaluate the Determinant of an n × n Matrix Apply Cramer’s Rule

Concept 1 Evaluate the Determinant of a 2 × 2 Matrix

Evaluate the Determinant of a 2 × 2 Matrix Associated with every square matrix is a real number called the determinant of the matrix. A determinant of a square matrix A, denoted |A|, is written by enclosing the elements of the matrix within two vertical bars. The determinant of the 2 × 2 matrix

Example 1 Evaluate the determinant of the matrix.

Example 2 Evaluate the determinant of the matrix.

Skill Practice 1 Evaluate the determinant.

Concept 2 Evaluate the Determinant of an n × n Matrix

Evaluate the Determinant of an n × n Matrix (1 of 10) For any element of that element is the determinant of the resulting matrix obtained by deleting the

Evaluate the Determinant of an n × n Matrix (2 of 10) Determinant of a 3  3 Matrix: or equivalently:

Evaluate the Determinant of an n × n Matrix (3 of 10) We can choose to find the determinant by expanding the minors of any row or column. However, we must choose the correct sign to apply to the product of factors of each term. The following array of signs is helpful. Notice that the sign is positive if the sum of the row and column number is even. The sign is negative if the sum of the row and column number is odd

Evaluate the Determinant of an n × n Matrix (4 of 10) To evaluate the determinant of an n  n matrix, choose any row or column. For each element in the selected row or column, multiply the minor by 1 or –1 depending on whether the sum of the row and column number is even or odd. That is, for the element This products is called the cofactor of the element

Evaluate the Determinant of an n × n Matrix (5 of 10) Example: 2 is in the 3rd row and 1st column. The cofactor of 2 is

Evaluate the Determinant of an n × n Matrix (6 of 10) Evaluating the Determinant of an n  n Matrix by Expanding Cofactors Step 1: Choose any row or column. Step 2: Multiply each element in the selected row or column by its cofactor. Step 3: The value of the determinant is the sum of the products from Step 2.

Evaluate the Determinant of an n × n Matrix (7 of 10) The determinant of a 3 ×3 matrix can also be evaluated by using the “method of diagonals.” This method applies only to the determinant of a 3 ×3 matrix. Example: Step 1: Recopy columns 1 and 2 to the right of the matrix

Evaluate the Determinant of an n × n Matrix (8 of 10) Step 2: Multiply the elements on the diagonals beginning in the upper left corner. Then multiply the three diagonals in the other direction beginning in the upper right corner. (Each diagonal has 3 elements)

Evaluate the Determinant of an n × n Matrix (9 of 10) Step 3: The value of the determinant is found by adding the first three products and subtracting the last three products.

Evaluate the Determinant of an n × n Matrix (10 of 10) Is a matrix is invertible? Let A be an n × n matrix. Then A is invertible if and only if A ≠ 0.

Example 3 Find the minor for each element of the first row.

Skill Practice 2 Find the minor for each element in the row of the matrix form Example 2.

Example 4 Evaluate the determinant of the matrix.

Skill Practice 3 Evaluate the determinant.

Example 5 Evaluate the determinant of the matrix.

Skill Practice 4 Evaluate the determinant by expanding the factors about the elements in the third column.

Example 6 Evaluate the determinant of the matrix and state whether the matrix is invertible.

Example 7 Evaluate the determinant of the matrix and state whether the matrix is invertible.

Skill Practice 5 Use |A| to determine if A is invertible.

Concept 3 Apply Cramer’s Rule

Apply Cramer’s Rule (1 of 5) Cramer’s rule involves finding the ratio of several determinants derived from the coefficients of the equations within the system.

Apply Cramer’s Rule (2 of 5) Cramer’s Rule for a System of Two Linear Equations in Two Variables Given the system Then if D ≠ 0, the system has the unique solution:

Apply Cramer’s Rule (3 of 5) For a system of n variables and n equations, Cramer's rule follows the pattern above. D = determinant of the matrix formed by the coefficients of x, y, z, w, etc. (D ≠ 0) determinant of the matrix formed by replacing the x-coefficient column in D with the constants on the right. determinant of the matrix formed by replacing the y-coefficient column in D with the constants on the right. 

Apply Cramer’s Rule (4 of 5) determinant of the matrix formed by replacing the z-coefficient column in D with the constants on the right. … and so on.

Apply Cramer’s Rule (5 of 5) Consider an n  n system of linear equations in which Cramer’s rule is used to find the solution. If D = 0 and at least one of the determinants in the numerator is nonzero, then the system has no solution. If D = 0 and all determinants in the numerator are zero, then the system has infinitely many solutions.

Example 8 Solve the system using Cramer's Rule. 3x - 5y = 7 12x + 3y = 5

Skill Practice 6 Solve the system by using the Cramer’s rule. 3x - 4y = 9 -5x + 6y = 2

Example 9 Use Cramer's Rule to solve for y. x + y + z = 5 2x - 3y - z = 40 x – y + z = 7

Skill Practice 7 Solve the system by using 5x + 3y - 3z = -14 3x - 4y + z = 2 x + 7y + z = 6

Example 10 Solve the system if possible by using Cramer’s rule. If the system does not have a unique solution, determine the number of solutions. 12x + 15y = -8 -4x - 5y = 10

Skill Practice 8 Solve the system by using Cramer’s rule if possible Otherwise, use a different method. x + 4y = 2 3x + 12y = 4