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 asdet or A determinant determines or identifies something.Well see exactly what this is shortly. 2x2 Determinant The determinant of a 22 matrix is the difference of the products of the diagonal elements: major minor. Exercise 2 Finddet . =3 2 6 1 Exercise 3 Evaluate the determinant of each matrix. 2 3 1 4 1 2 2 4
/5 Exercise 4 Find if =17. =3 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 Finddet . =4 1 2 3 2 Determinants (3x3) Expansion by Minors:
An alternative method involves finding the determinant in terms of three 22 matrices. Exercise 6 Find . = 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 1 , 1, 2 , 2, and 3 , 3 is given by: Exercise 7 Find the area of ABC. Objective 3 You will be able to use Cramers Rule to solve a linear system in 2 or 3 variables Exercise 8 Solve the matrix equation. 3 =15 13 Exercise 8 Solve the matrix equation. 3 4 2 5 = 15 13 34=15
3 =15 13 34=15 2+5= 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 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 Slope of : = Slope of : =
For this system to be either consistent and dependent or inconsistent, the equations must have the same slope: Slope of : = Determinant = 0 Slope of : = =0 Exercise 9 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. 34=15 2+5= 13 Cramers Rule (2x2) Mr. Cramer tells us that we can use determinants to solve a linear system. No elimination! No substitution! Gabriel Cramer (1750ish) Cramers Rule (2x2) Let be the coefficient matrix for the system: If det 0 , then the system has one solution, and Cramers Rule (2x2) When using Cramers rule, notice that the -value comes from replacing coefficients of with the constant terms.Likewise for the -value. Exercise 11 Solve the system using Cramers Rule. 34=15 2+5= 13 Exercise 12 Solve the system using Cramers Rule. 4+7=2 32=8 Cramers Rule (3x3) Exercise 13 Solve the system using Cramers Rule. 34+2=18
4+5=13 23+=11 Exercise 14 Solve the system using Cramers Rule. 2+3+=1 3+3+=1
2+4+=2 Determinants & Cramers Rule
3.7: Determinants and Cramer's Rule Determinants & Cramers 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 Cramers Rule to solve linear system in 2 or 3 variables