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Matrices: What Are They?

10th Grade Algebra II Honors Project
by

Megan Dempsey

on 27 May 2014

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Transcript of Matrices: What Are They?

Scalar Multiplication
Addition and Subtraction
Transpose of a Matrix
Matrix Multiplication
Matrix Operations
Megan Caroline Dempsey - Honors
Matrices: What Are They?
What Are Matrices?
3
2
6
5
-2
-7
-10
8
3
Rectangular array of numbers
Named by dimensions; ex. 2x3, 3x3, 2x4, etc.
Way of representing information:
Computer graphics/science (numbers represent other things, ex. color intensity, objects, movement. etc.)
Physics (quantum mechanics, electrical circuits, optics, etc.)
Stochastic matrices and Eigen vector solvers used in ranking of Google search.
7
5
-10
3
8
0
3
x
Scalar
=
21
15
-30
9
24
0
-1 -7 5
0 3 -10
+
5 0 3
11 -1 7
=
6 -7 8
11 2 -3
0 1
3 2
-
-1 3
0 5
=
1
-2
3
-3
Different Dimensions
3 2 0
4 6 7
+
5 7

-1 6
2 x 2
2 x 3
=
Undefined
Cannot + or - matrices of different dimensions, just like you cannot add, subtract, multiply or divide roots of different #'s.
3

2xyz

3xyz
1 -2
3 0
7 5
A =
A
T
=
transpose of A
Transpose = rows become columns
columns become rows
A
T
=
1 3 7
-2 0 5
2
-2
5
3
X
-1
4
7
-6
2(-1) + -2 (7)
2(4) + -2(-6)
5(-1) + 2(7)
5(4) + 3(-6)
-16 20

9 2
2 x 3 times 3 x 2
column
row
Order matters: ED ≠ DE, i.e. Matrix multiplication is not commutative
Properties of Matrix Multiplication
Is Associative: A(BC) = (AB)C
Identity Matrix
1 x a = a

1 x b = b

1 x 5 = 5
1 = identity for scalar numbers
2 X 2 Matrix:
1 0

0 1
3 x 3 Matrix:
1 0 0

0 1 0

0 0 1
4 X 4 Matrix:
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
Inverting a 2x2 Matrix
finding the determinant
Inverse 2 x 2:
1
determinant
x
Adjugate
Example: Matrix A
A =
5 3
-1 4
determinant =
5 3
-1 4
5(4) - 3(-1)
Determinant of Matrix A =
23
Adjugate
Example Matrix A:
Matrix A =
3 5

-7 2
Adjugate =
3 5

-7 2
(-)
(-)
Adjugate of A =
2 -5

7 3
3 5

-7 2
Inverse of:
determinant = 41
1
41
x
2 -5

7 3
Inverse =
2
41
-5
41
7
41
3
41
A x A
-1
=
Identity Matrix
Systems of Equations
3x + 2y = 7
-6x + 6y = 6
=
3 2
-6 6
x
y
=
7
6
A
b
c
Ab = c
multiply everything by the inverse of A
A
-1
Ab
=
A
-1
c
Ib =
A
-1
c
b =
A
-1
c
Inverting a 3x3 Matrix
Finding the determinant
Method 1:
Example Matrix A:
4 -1 1
4 5 3
-2 0 0
4 -1 1
4 5 3
-2 0 0
4
4
-2
-1
5
0

+
-
4(5)(0) + -1(3)(-2) + 1(4)(0)
-1(4)(0) - 4(3)(0) - 1(5)(-2)
0
6
0
0
0
10
10 + 6 = 16
Method 2:
4 -1 1
4 5 3
-2 0 0
4 -1 1
4 5 3
-2 0 0
+ - +
+ 4
5 3
0 0
-
-1
4 3
-2 0
+
1
4 5
-2 0
0
6
10
10 + 6 = 16
Example Matrix C=
-1 -2 2
2 1 1
3 4 5
Matrix of Minors
-1 -2 2
2 1 1
3 4 5
=
1 1
4 5
2 1
3 5
2 1
3 4
-2 2
4 5
-1 2
3 5
-1 - 2
3 4
-2 2
1 1
-1 2
2 1
-1 -2
2 1
1 7 5
-18 -11 2
-4 -5 3
Cofactor Matrix
+ - +
- + -
+ - +
1 7 5
-18 -11 2
-4 -5 3
1 -7 5
18 -11 -2
-4 5 3
Determinant: 23
1
23
Cofactor matrix
T
1
23
1 18 -4
-7 -11 5
5 -2 3
1
23
18
23
-4
23
-7
23
-11
23
5
23
5
23
-2
23
3
23
Augmented Matrix
2x - 3y = 8
4x + 5y = 1
2
4
-3
5
8
1
Reduced Row Echelon
Used to find solutions for systems and tell whether it is dependent or inconsistent.
Rules:
Interchange rows
Multiply entries by a nonzero #
Add constant times row to another.
2
4
-3
5
8
1
(1/2)
1
4
-1.5
5
4
1
-4
(+)
1
0
-1.5
11
4
-15
(1/11)
1
0
-1.5
1
4
-15/11
(3/2)
(+)
1
0
0
1
43/22
-15/11
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