Present Remotely
Send the link below via email or IM
CopyPresent to your audience
Start remote presentation Invited audience members will follow you as you navigate and present
 People invited to a presentation do not need a Prezi account
 This link expires 10 minutes after you close the presentation
 A maximum of 30 users can follow your presentation
 Learn more about this feature in our knowledge base article
Do you really want to delete this prezi?
Neither you, nor the coeditors you shared it with will be able to recover it again.
Make your likes visible on Facebook?
Connect your Facebook account to Prezi and let your likes appear on your timeline.
You can change this under Settings & Account at any time.
Matrices
No description
by
TweetSofia Baeta
on 27 May 2014Transcript of Matrices
Matrices
First of ALL
WHAT IS MATRICES?
Adding and Subtracting
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
THE ANSWER GOES HERE
=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)
.
adjunct (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
adj (C) =
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
Full transcriptFirst of ALL
WHAT IS MATRICES?
Adding and Subtracting
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
THE ANSWER GOES HERE
=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)
.
adjunct (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
adj (C) =
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