MATRIX 1

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

NOTATION OF MATRIX

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

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

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

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

3. Determine array elements of the 2nd row?

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

TYPES OF MATRIX

ROW MATRIX

COLUMN MATRIX

DIAGONAL MATRIX

IDENTITY MATRIX Addition Multiplication Zero Matrix

TRIANGLE MATRIX Upper Triangle Atas Lower Triangle

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

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

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

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

2 = 2  6 = 2x x = 6/2 = 3  x + y = -5 3 + y = -5 y = -5 - 3 = -8  z = 4x – y = 4.3 – (-8) z = 12 + 8 = 20

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

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

Contoh: Jika Dapatkah A dan C dijumlahkan?

If given A + B = … B - A = …

2. MULTIPLICATION OF MATRIX a. Multiplication two matrices =

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

If given A x C = … A 3 x 2 C2 x 4 =

B1A B2A B3A K1C K2C K3C K4C

B1A B2A B3A K1C K2C K3C K4C a = (6x3)+(2x4) = 18 + 8 = 26

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

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

26 B1A B2A B3A -9 25 K1C K2C K3C K4C

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

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]

Example Determine 2 x A if

Answer 2.A = =

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

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

Determine value of determinant matrix below Example Determine value of determinant matrix below Answer: |A| = 5.6 – 10.-1 = 30 + 10 = 40

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

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

Determine determinat of Example Determine determinat of Answer: = (0.1.5 + 4.-3.-1 + 7.2.3) –(-1.1.7 +3.-3.0 + 5.2.4) = (0+12+42) – (-7+0+40) = 54 – 33 = 21

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

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

Kofactor If than kofactor M11 = 6 .-11+1 = 6 M12 = -1. -11+2 = -1.-1 =1 M21 = 10. -12+1 = 10. -1 = -10 M22 = 5. -1 2+2 = 5.1 = 5

Adjoin If than Adjoin matrix A Resulted from the its kofactor

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

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

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

Kofactor

Adjoin

5. INVERSE Inverse matrix A 

a. Inverse ordo 2 X 2

Determine inverse from Contoh: Determine inverse from Answer

Answer :

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

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

Competence Check Given (A.B)-1 = ….

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