Download Chapter 8 Matrices and Determinants

Chapter 8
Matrices and Determinants
By Richard Warner, Nate Huyser,
Anastasia Sanderson, Bailey Grote
Chapter 8.1: General Matrices
• Rectangular array of numbers called entries
• Dimensions of a matrix are number of rows by
the number of columns
 A11
 A21

 A31
A12
A22
A32
A13

A23
A33
Chapter 8.1: Augmented Matrices
• Augmented Matrix- derived from a system of
equations
• Elementary Row Operations
• Interchange any two rows
• Multiply any row by a nonzero constant
• Add two rows together
• 2x2 by hand, 3x3 with calculator
Chapter 8.1: Reduced Row Echelon
Form (RREF)
• Any rows consisting of all zeros occur at the bottom
of the matrix
• All entries on the main diagonal are 1
• All entries not on the main diagonal or in the last
column are 0
• A13 is the x-coordinate of the solution
• A23 is the y-coordinate of the solution
1
0

0
1
x

y
Chapter 8.1: Gauss Jordan Elimination
• Uses Augmented Matrices to solve systems of
equations
1. Write system as an augmented matrix
2. Use the row operations to make A11 = 1
3. Work down, around, and up to achieve RREF
4. Write last column as ordered pair for final answer
Chapter 8.1: Solving with Calculator
(RREF)
•
1.
2.
3.
4.
5.
6.
7.
Only used for Matrices larger than 2x2
(2nd) [Matrix] → EDIT
1 0 0
Matrix[A] 3x4
Enter entries by rows 0 1 0

(2nd) [Quit]
(2nd) [Matrix] → MATH 0 0 1
Select [RREF]
(2nd) [Matrix] select Martix[A]
x

y
z 
Chapter 8.2: Matrix Operations
Equality of Matrices: 2 matrices are equal if
they have the same dimensions and their
corresponding entries are equal
To add and subtract Matrices: They must
have the same dimensions.
•Add the corresponding entries
Scalar Multiplication:
•Multiplying a matrix by a scalar (constant)
•Multiply each entry in the matrix by the scalar
Chapter 8.2: Matrix Operations
Matrix Multiplication:
• To Multiply AB, A’s columns must equal B’s
rows
• Multiply the entries in A’s rows by the
corresponding entries in B’s columns
• Amxn* Bnxr =ABmxr
Ex: p.598 #29
8.3 Inverse Matrices
Identity Matrices
I 2x2
1 0
0 1 


I 3x3
1 0 0
0 1 0 


0 0 1
Cont.
8.3 Inverse Multiplication
Inverse of2x2:
• A A-1 =A-1 A =I
• If A= a b  where ad-bc cannot equal 0,
c
d
Then A-1 =1/(ad-bc) *
 d  b
 c a 


Cont.
8.3 Inverse Multiplication
Inverse of 3x3
1. Enter [matrix] in calculator
2. [matrix][A] [enter] [x-1 ] [enter]
To solve a system of linear equations
1. Write the system of equations as a matrix problem
2. Find A-1
3. X=A1B
 x1

 x2
 x3
y1
y2
y3
z1 
x
z2 
z 3 
 x
 y
 
 z 
=
 c1 
c 
 2

 c3 

8.4 Determinants
• a real number derived from a square matrix
• If A = CA DB  then Det[A]= AD-CB


• For 2x2 matrices only
• For 3x3 matrices or larger
1.
2.
3.
4.
5.
(2nd) Matrix → [Edit] A
Enter dimensions
(2nd) Quit
(2nd) Matrix → [Math] enter
(2nd) Matrix → enter
8.5 Determinant Applications
• Cramer’s Rule solves systems using
determinates.
Dy
Dx
x
• Example:
D
y
e b
f d
x
a b
c d
D
a e
c f
y
a b
c d
8.5 Determinant Applications
• Finding the area of
a triangle where
the points are (a,b),
(c,d), (e,f)
a
1
 c
2
e
b 1
d 1A
f 1
• Points are collinear
if A=0
a
c
e
b 1
d 1 0
f 1