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
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