2. AccPeCalc Matrices review

5. Definition of an Inverse Given a n x n matrix A, if there exists an inverse (A -1 ) of matrix A then A A -1 = A -1 A = I n Example: A A -1 = 1 2 7 9 -1.5.5 1.25 -.25 1 0 0 1

6. If a square matrix A has NO inverse then it is called a singular matrix. If a square matrix has an inverse it is called a nonsingular matrix.

7. Example Verifying an inverse matrix Prove that A = and B = Are inverse matrices: 3-2 -1 1 1 2 1 3

8. Finding a determinant of square matrices Determinant of a 2 x 2 If a 2 x 2 matrix A has an inverse then the determinant of A is as follows. A =, det A = ad - bc a b c d

9. Using determinants to find the area of a triangle A triangle with vertices at (x 1, y 1 ), (x 2,y 2 ), and (x 3,y 3 ) Use: Area of the triangle = ½ base x height = ½ ( det ) X 1 y 1 1 X 2 y 2 1 X 3 y 3 1

10. Find the area of the following triangle The triangle with the following vertices: (4,0), (7,2), and (2,3)

11. Solving systems of two equations. An example of a system of two equations is as follows: X + y = 3 X – 2y = 0 We have solved systems of two equations in the past graphically, algebraically, and using the method of substitution. In this lesson we will review these methods. Notice we have 2 equations with 2 unknowns.

12. Systems of equations A solution of a system of two equations in two variables is an ordered pair of real numbers that is a solution to each equation. The solution to this graph is the ordered pair (3,4)

13. X + y = 3 Intersecting lines x – 2y = 0 Step 1: Equation 1: Y = -x + 3 Equation 2: -2y = -x y = ½ x Remember: the number in front of the x is the slope. Solution is (2,1) x y

14. Y =2x – 1 Parallel lines y = 2x + 2 These lines never intersect! Since the lines never cross, there is NO SOLUTION! Parallel lines have the same slope with different y- intercepts.

15. Coinciding lines (lines that lay on top of each other) These lines are the same! Since the lines are on top of each other, there are INFINITELY MANY SOLUTIONS! Coinciding lines have the same slope and y-intercepts.

16. What is the solution of the system graphed below? 1.(2, -2) 2.(-2, 2) 3.No solution 4.Infinitely many solutions