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Matrices

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by

Sofia Baeta

on 27 May 2014

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Transcript of Matrices

Matrices
First of ALL
WHAT IS MATRICES?
Multiplication
Sofia Baeta
A Matrix is an array of numbers:
6 4 24
1 -9 8

There are many things we can do with them ...

2 rows
3 columns
2x3 matrix
1
3 5
0 0
-2 -7
2 5 8
0 -3 11
1x1
3x2
2x3
"tagging" system:
5 4
-2 11
0 3
A=
a
i j
i j
th th
ROW
COLUMN
i
j
5 4
-2 11
0 3
Example:
a a
a a
a a
1 1
1 2
2 1
2 2
3 1
3 2
is entry
is entry
is entry
is entry
is entry
is entry
5
-2
0
4
11
3
a
a
a
a
a
a
1 1

2 1

3 1

1 2

2 2

3 2
+
A
B
=
C
=
+
1
6
3
-3
4
11
-7
0
0
5
5
-4
7
0
15
-7
5
1
1+6=
3+(-3)=
A
B
C
-
=
=
-
-2
7
0
-8
3
0
-9
8
3
(-2)-(7)=
+
=
THE SIZES MUST BE THE SAME!
x
=
2 -2
5 3
-1 4
7 -6
(2)*(-1) + (-2)*(7)
(5)*(-1) + (3)*(7)
(2)*(4) + (-2)*(-6)
(5)*(4) + (3)*(-6)
-16
20
16
2
=A
B=
0 9
8 -6
4 7
-1 5
2 3
B*A
=C
C
1 1
C
1 2
C
C
C
C
2 1
2 2
3 1
3 2
18
0
18 -27

-20 58

10 -1
B x A= C
3x2
2x2
3x2
A x B
2x2
3x2
0 9
8 -6
4 7
-1 5
2 -3
There's no partner for 4
the order of the factors alter the product
undefined
undefined
Division

Inversion
Identity Matrix
1 x 1 = 1
1 x 2 = 2
1 x 3 = 3
1 x 4 = 4
1 x (anything) = anything
Is there some MATRIX that has this property ?
YES !
1 =
= Identity Matrix
1 x = x
.
=
.
IM
A
A
3x3
1 2 3
4 5 6
7 8 9
3x3
3x3
1 2 3
4 5 6
7 8 9
1 0 0
0 1 0
1 0 0
3x3
I
4x4
2x2
1 0
0 1

1 0 0
0 1 0
0 0 1

1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1

A=
3 5
-7 2
=
1
determinant (A)
.
B=
5 3
-1 4

5
3
-1
4
=
(5x4)
-

(3x(-1))
20

+
3
23
A B
C D
=
(A*D)-(B*C)
1
(3*2)-(-7)*5

3

5
-7

2
2

-5

7

3
-1
-1
2 -5
7 3
6+35
41
2 -5
7 3
X
=
2/41 -5/41
7/41 3/41
Matrices to solve a system of equtions
What a matrix is actually good for
ALL a matrix is a way of representing data
3x + 2y = 7
6x + 6y = 6
3 2
-6 6
x
y
.
=
7
6
ax= b
x= b/a
multiply by the inverse
X
inverse
A
-1
=
1
A
.
6 -2
6 3
18- (-12) = 30
A
-1
=
6 -2
6 3
1
30
x
y
=
1
30
6 -2
6 3
7
6
result/ 30 = x and y
Finding the determinant of a 3x3
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
+ +
-
0 6 0
-10 0 0
- -
16
Inverting 3x3
C=
-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*5)-(1*4)= 1
1 7 5
-18 -11 2
-4 -5 3

+ - +
- + -
+ - +

det (C) =
-1 -2 2
2 1 1
3 4 5
-1 -2
2 1
3 4
=
-5 -6 +16 - -20 - -4 -6
+5 +20 + 4 -6
= 23
C =
-1
1
23
1
23
Transpose
1 -7 5
18 -11 -2
-4 5 3

1
18
-4
-7
-11
5
5
5
3

1 18 -4

-7 -11 5

5 5 3
23
23
23
23
23
23
23
23
23
Brief
Augmented Matrix
Matrix Coding
Reduce row echelon
Game theory
Economics
Chemistry
Quantum Theory
Computer Graphis
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