4.3 Determinants and Cramer’s rule How do you find the determinant of a matrix? How do you find the area of a triangle given 3 sets of coordinates? How is Cramer’s Rule used to solve a 2 by 2 system? A 3 by 3 system?

4.3 Determinants Associated with every square matrix is a whole number called the determinant The Determinant of a Matrix A is denoted by detA or |A|

= ad - cb Determinant of a 2x2

Ex 8 Evaluate the determinant =1(7)— 2(4) = = -1

Ex 9 Evaluate the determinant =7(3)— 2(2) = = 17

Ex 10 Evaluate = -73 Step 1: recopy the first two columns = (2*0*6+ -3*3*1+ 4*-2*2) ---- (1*0*4 + 2*3*2 + 6*-2*-3) = ( ) – ( ) = Step 2: multiply the down diagonals and add the products. Step 3: multiply the up diagonals and add the products NOTE: You subtract the up diagonal from the down diagonal

Ex 11 Evaluate. = -89 det

Area of a Triangle! The area of a triangle with vertices (x 1,y 1 ), (x 2,y 2 ), and (x 3,y 3 ) Area = *Where ± is used to produce a positive area!!

Find the area of the triangle. Area= = Square units

Find the area of the triangle. Area= = Square units

Cramer’s Rule (because Cramer RULES!) Gabriel Cramer was a Swiss mathematician ( )

Coefficient Matrices You can use determinants to solve a system of linear equations. You use the coefficient matrix of the linear system. Linear SystemCoeff Matrix ax+by=e cx+dy=f

Cramer’s Rule for 2x2 System Let A be the coefficient matrix Linear SystemCoeff Matrix ax+by=e cx+dy=f If detA ≠ 0, then the system has exactly one solution: and

Example 1- Cramer’s Rule 2x2 Solve the system: 8x+5y=2 2x-4y=-10 The coefficient matrix is: and So: and

Solution: (-1,2)

Example 2- Cramer’s Rule 2x2 Solve the system: 2x+y=1 3x-2y=-23 The solution is: (-3,7) !!!

Example 3- Cramer’s Rule 3x3 Solve the system: x+3y-z=1 -2x-6y+z=-3 3x+5y-2z=4 Let’s solve for Z Z=1 The answer is: (2,0,1)!!!

Assignment p. 218, odd