AccPeCalc Matrices review Definition of an Inverse Given a n x n matrix A, if there exists an...
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Transcript of AccPeCalc Matrices review Definition of an Inverse Given a n x n matrix A, if there exists an...
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