1. MATRIX 1

2. DEFINITION
Matrix is rectangular array of numbers, consists of rows and columns and is written using brackets or parentheses.

3. NOTATION OF MATRIX

4. MATRIX ELEMENTS
1st row elements
Elements position
1st Column elements

5. ORDO
Ordo m x n
Notation : A m x n
1st row 1
2nd row
mth row
1st Column
2st Column
3rd Column
nth Column

6. Example:
1. What is the name of matrix above?

7. 2. Determine the array element of 3th row and 4th
column!

8. 3. Determine array elements of the 2nd row?

9. 3. Determine dimension of Matrix Z!
Ordo 3 x 4
Notation Z 3 x 4

10. TYPES OF MATRIX

11. ROW MATRIX

12. COLUMN MATRIX

13. DIAGONAL MATRIX

14. IDENTITY MATRIX
Addition
Multiplication
Zero Matrix

15. TRIANGLE MATRIX
Upper Triangle Atas
Lower Triangle

16. MATRIX TRANSPOSE
Transpose matriks A happened if each elements of row matrix A change be come element of column of matrix A’, so A m x n become A’ n x m.
1st row elements A  1st column elements A’,
2nd row elements A  2nd column elements A’,
etc

17. TRANSPOSE of MATRIX
A’ 2 x 4
A 4 x 2 

18. SIMILARITY OF TWO MATRIX
Giveb
If A = B, determine the value of x, y dan z!

19. 2 = 2
x + y = -5
6 = 2x
z = 4x - y

20. 2 = 2
 6 = 2x
x = 6/2 = 3
 x + y = -5
3 + y = -5
y = = -8
 z = 4x – y
= 4.3 – (-8)
z = = 20

21. -5
3 + (-8) 20
2.3
4.3 – (-8)
12 + 8 6 20

22. 1. ADDITION AND SUBSTRACTION OF MATRIX
Two or more matriks can be addition or subtraction if :
Both of them have same Ordo
Operated only for elements in the similar position

23. Contoh:
Jika
Dapatkah A dan C dijumlahkan?

24. If given
A + B = …
B - A = …

25. 2. MULTIPLICATION OF MATRIX
a. Multiplication two matrices =

26. Example
If given
Can A multiplication with C?
A 3 x 2 C2 x 4 =

27. If given
A x C = …
A 3 x 2 C2 x 4 =

28. B1A
B2A
B3A
K1C K2C K3C K4C

29. B1A
B2A
B3A
K1C K2C K3C K4C
a = (6x3)+(2x4) =
= 26

30. 26
B1A
B2A
B3A
K1C K2C K3C K4C
a = (-3x3)+(0x4) =
= -9

31. 26
B1A
B2A
B3A -9
K1C K2C K3C K4C
a = (5x5)+(0x-8) =
= 25

32. 26
B1A
B2A
B3A -9 25
K1C K2C K3C K4C

33. 26 1 -4 30 -9
-1,5
-15
A.C =
-17
10,5 16 25

34. b. Scale multiplication with matrix
Multiplication a real number with matrix A is multipilcation each elements of matrix A by that real number
k.A = [k.amn]

35. Example
Determine 2 x A if

36. Answer
2.A = =

37. DETERMINANT
Determinant of matrix
Only used in square
are substraction with elements 1st diagonal and 2nd diagonal, where each elements enclosed

38. a. DETERMINANT ORDO 2 X 2 If
than|A| = ad - bc

39. Determine value of determinant matrix below
Example
Determine value of determinant matrix below
Answer:
|A| = 5.6 – = = 40

40. DETERMINAN ORDO 3 x 3
If given
than |A| =

41. DETERMINAN ORDO 3 x 3
|A| =
= (a.e.i + b.f.g + c.d.h) –(c.e.g + a.f.h + b.d.i)

42. Determine determinat of
Example
Determine determinat of
Answer:
= ( ) –( )
= ( ) – ( )
= 54 – 33 = 21

43. 4. ADJOIN
Adjoin matrix A is the result transpose from kofaktor matriks A.
Matrix A
Minor Matrix A
Kofaktor Matrix A
Adjoin Matrix A

44. a. Ordo 2 x 2
Minor
Jika maka minor
M12 = -1
M11 = 6
M21 = 10
M22 = 5

45. Kofactor
If than kofactor
M11 = = 6
M12 = = =1
M21 = = = -10
M22 = = 5.1 = 5

46. Adjoin
If than Adjoin matrix A
Resulted from the its kofactor

47. b. Ordo 3 x 3
If , minor matrix A showed
next
M11 = 2.-2 – (0.3)
= -4- 0
= -4
M12 = 1.-2 – (-5.3)
= -2 – (-15)
= 13
M13 = 1.0 – (-5.2)
= 0 – (-10)
= 10

48. M21 = 1.-2 – 0.2
= -2- 0
= -2
M22 = 1.-2 – (-5.2)
= -2 – (-10)
= 8
M23 = 1.0 – (-5.1)
= 0 – (-5)
= 5

49. M21 = 1.3 – (2.2)
= 3 - 4
= -1
M22 = 1.3 – (1.2)
= 3 – 2
= 1
M23 = 1.2 – (1.1)
= 2 – (1)
= 1

50. Kofactor

51. Adjoin

52. 5. INVERSE
Inverse matrix A 

53. a. Inverse ordo 2 X 2

54. Determine inverse from
Contoh:
Determine inverse from
Answer

55. Answer :

57. II. MATRIX APPLICATION
Using to determine variabel value of linear equation. If the equation have variabel x dan y, than ..

58. Example Determine value of x dan y from the next equations 2x + 3y = 7

60. Competence Check
Given
(A.B)-1 = ….

61. 2. Determine solution set from the next l
are ….