Objective 1 You will be able to find the determinant of a 2x2 and a 3x3 matrix

Determinant The determinant of a square matrix is simply a real number that is associated with that matrix. Denoted as det 𝐴 or 𝐴 A determinant determines or identifies something. We’ll see exactly what this is shortly.

2x2 Determinant The determinant of a 2×2 matrix is the difference of the products of the diagonal elements: major – minor.

Exercise 2 Find det 𝐴 . 𝐴= 3 −2 6 1

Exercise 3 Evaluate the determinant of each matrix. 2 −3 1 4 −1 2 2 −4 1 3 2 2/5

Exercise 4 Find 𝑥 if 𝐴 =17. 𝐴= 3 𝑥 1 5

3x3 Determinant Diagonals Method: This one is a little crazier. First you have to duplicate the first two columns and then write them after the last column.

3x3 Determinant Diagonals Method: Now find the sum of the products of the major diagonal elements. Finally subtract the sum of the products of the minor diagonal elements.

Exercise 5 Find det 𝐴 . 𝐴= 4 −1 2 −3 −2 −1 0 5 1

Determinants (3x3) Expansion by Minors: An alternative method involves finding the determinant in terms of three 2×2 matrices.

Exercise 6 Find 𝐴 . 𝐴= −1 1 2 0 2 3 3 4 2

Objective 2 You will be able to find the area of a graphed triangle using determinants

Area of a Triangle Finding the area of a triangle in the coordinate plane is as easy as taking half of the determinant an augmented matrix.

Area of a Triangle The area of a triangle with vertices at 𝑥 1 , 𝑦 1 , 𝑥 2 , 𝑦 2 , and 𝑥 3 , 𝑦 3 is given by:

Exercise 7 Find the area of ABC.

Objective 3 You will be able to use Cramer’s Rule to solve a linear system in 2 or 3 variables

Exercise 8 Solve the matrix equation. 3 −4 2 5 𝑥 𝑦 = −15 13

Exercise 8 Solve the matrix equation. 3 −4 2 5 𝑥 𝑦 = −15 13 3𝑥−4𝑦=−15 3 −4 2 5 𝑥 𝑦 = −15 13 3𝑥−4𝑦=−15 2𝑥+5𝑦= 13 Coefficient Matrix

Coefficient Matrix A coefficient matrix is formed by arranging the coefficients of a linear system in a square array.

Determinants Determine A determinant in mathematics is a number that determines or identifies the nature of something. The determinant of a coefficient matrix determines if a linear system has a unique solution.

Exercise 9 𝐴 𝐵 Assume that the system above does not have a unique solution; that is, the system is either consistent and dependent or inconsistent. What must be true about the system? How does this relate to the determinant of the coefficient matrix for the system?

Exercise 9 − 𝑎 𝑏 Slope of 𝐴 : − 𝑎 𝑏 =− 𝑐 𝑑 − 𝑐 𝑑 Slope of 𝐵 : 𝑎𝑑=𝑏𝑐 For this system to be either consistent and dependent or inconsistent, the equations must have the same slope: − 𝑎 𝑏 Slope of 𝐴 : − 𝑎 𝑏 =− 𝑐 𝑑 Determinant = 0 − 𝑐 𝑑 Slope of 𝐵 : 𝑎𝑑=𝑏𝑐 𝑎𝑑−𝑏𝑐=0

Exercise 9 𝐴 𝐵 For this system to be either consistent and dependent or inconsistent, the equations must have the same slope: If the determinant of a coefficient matrix for a linear system is zero, then the system must be consistent and dependent (same line) or inconsistent (parallel).

Exercise 10 Determine if the system below is consistent and independent, consistent and dependent, or inconsistent. 3𝑥−4𝑦=−15 2𝑥+5𝑦= 13

Cramer’s Rule (2x2) Mr. Cramer tells us that we can use determinants to solve a linear system. No elimination! No substitution! Gabriel Cramer (1750ish)

Cramer’s Rule (2x2) Let 𝐴 be the coefficient matrix for the system: If det 𝐴≠0 , then the system has one solution, and

Cramer’s Rule (2x2) When using Cramer’s rule, notice that the 𝑥-value comes from replacing coefficients of 𝑥 with the constant terms. Likewise for the 𝑦-value.

Exercise 11 Solve the system using Cramer’s Rule. 3𝑥−4𝑦=−15 2𝑥+5𝑦= 13

Exercise 12 Solve the system using Cramer’s Rule. 4𝑥+7𝑦=2 −3𝑥−2𝑦=−8

Cramer’s Rule (3x3)

Exercise 13 Solve the system using Cramer’s Rule. 3𝑥−4𝑦+2𝑧=18 4𝑥+𝑦−5𝑧=−13 2𝑥−3𝑦+𝑧=11

Exercise 14 Solve the system using Cramer’s Rule. 2𝑥+3𝑦+𝑧=−1 3𝑥+3𝑦+𝑧=1 2𝑥+4𝑦+𝑧=−2

Determinants & Cramer’s Rule 3.7: Determinants and Cramer's Rule Determinants & Cramer’s Rule Objectives: To find the determinant of a 2x2 and a 3x3 matrix To find the area of a triangle in the coordinate plane using determinants To apply Cramer’s Rule to solve linear system in 2 or 3 variables