AccPeCalc Matrices review

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 = In

Example:

A A -1 = A A-1 = 1 2

7 9

-1.5 .5

1.25 -.25

1 0

0 1

• 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.

Example Verifying an inverse matrix

• Prove that A = and B =

Are inverse matrices:

3 -2

-1 1

1 2

1 3

Finding a determinant of square matrices

Determinant of a 2 x 2If 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

Using determinants to find the area of a triangle

A triangle with vertices at (x1, y1), (x2,y2), and (x3,y3)

Use: Area of the triangle = ½ base x height =

½ ( det ) X1 y1 1

X2 y2 1

X3 y3 1

Find the area of the following triangle

The triangle with the following vertices: (4,0), (7,2), and (2,3)

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.

Systems of equationsA 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)

X + y = 3 Intersecting lines x – 2y = 0 Step 1: Equation 1: Y = -x + 3Equation 2: -2y = -x y = ½ xRemember: the number in front of the x is the

slope.

Solution is (2,1)

x

y

Y =2x – 1 Parallel linesy = 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.

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.

What is the solution of the system graphed below?

1. (2, -2)2. (-2, 2)3. No solution4. Infinitely many solutions