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Konsep Matriks

MATRIKS. Konsep Matriks. MATRIX. Concept of Matrix. Macam-macam Matriks. Kompetensi Dasar : Mendeskripsikan macam-macam matriks Indikator : Matriks ditentukan unsur dan notasinya Matriks dibedakan menurut jenis dan relasinya. Kinds of Matrix. Basic Competences :

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Konsep Matriks

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  1. MATRIKS Konsep Matriks

  2. MATRIX Concept of Matrix

  3. Macam-macam Matriks Kompetensi Dasar : Mendeskripsikan macam-macam matriks Indikator : • Matriks ditentukan unsur dan notasinya • Matriks dibedakan menurut jenis dan relasinya Matriks

  4. Kinds of Matrix Basic Competences : Describing the kinds of matrix Indicators : • Matrix is determined by its elements and notations • Matriks matrix is distinguished by its kinds and relations Matriks

  5. Macam – macam Matriks Pengertian Matriks • Matriks adalah susunan bilangan-bilangan yang terdiri atas baris-baris dan kolom-kolom. • Masing-masing bilangan dalam matriks disebut entri atau elemen. Ordo (ukuran) matriks adalah jumlah baris kali jumlah kolom. a11 a12…….a1j ……a1n a21 a22 ……a2j…….a2n : : : : ai1 ai2 ……aij…….. ain : : : : am1 am2……amj……. amn baris A = Notasi: Matriks: A = [aij] Elemen: (A)ij = aij Ordo A: m x n kolom Matriks

  6. Kinds of Matrix Definition of Matrix • Matrix is the arrangement of numbers which consists of rows and columns. • Each of the numbers in matrix is called as entry or element. Order (size) of matrix is the value of the row number multiplied by the number of column. a11 a12…….a1j ……a1n a21 a22 ……a2j…….a2n : : : : ai1 ai2 ……aij…….. ain : : : : am1 am2……amj……. amn rows A = Notation: Matrix: A = [aij] Element: (A)ij = aij Order A: m x n column Matriks

  7. 2 5 1 -8 25 -2 0 14 8 Macam-macam Matriks 1. Matriks Baris Matriks baris adalah matriks yang hanya terdiri dari satu baris. Matriks

  8. 2 5 1 -8 25 -2 0 14 8 Kinds of Matrix 1. Row matrix Row matrix is a matrix which consists of one row. Matriks

  9. 2 -7 9 2 1 Macam-macam Matriks 2. Matriks Kolom Matriks Kolom adalah matriks yang hanya terdiri dari satu kolom Matriks

  10. 2 -7 9 2 1 Kinds of Matrix 2. Column matrix Column matrix is a matrix which consists of one column. Matriks

  11. Macam – macam Matriks 3. Matriks Persegi Matriks persegi (bujur sangkar) adalah matriks yang jumlah baris dan jumlah kolom sama. 1 2 4 2 2 2 3 3 3 Trace(A) = 1 + 2 + 3 diagonal utama Trace dari matriks adalah jumlahan elemen-elemen diagonal utama Matriks

  12. Kinds of Matrix 3. Square matrix Square matrix is a matrix which has the same numbers of rows and columns. 1 2 4 2 2 2 3 3 3 Trace(A) = 1 + 2 + 3 Main diagonal Trace from matrix is the total numbers from the main diagonal elements. Matriks

  13. Macam- macam Matriks 4. Matriks Nol Matriks nol adalah matriks yang semua elemennya nol 0 0 0 0 0 0 0 Matriks identitas adalah matriks persegi yang elemen diagonal utamanya 1 dan elemen lainnya 0 I3 I4 I2 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 1 0 0 1 Matriks

  14. Kinds of Matrix 4. Zero matrix zero matrix is a matrix which all of its elements are zero. 0 0 0 0 0 0 0 Matrix identity is a square matrix which its main diagonal element is 1 and the other element is 0. I3 I4 I2 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 1 0 0 1 Matriks

  15. 0 -1 1 0 0 1 -1 0 A = AT= ½√2 -½√2 ½√2 ½√2 ½√2 ½√2 -½√2 ½√2 B = BT= Macam-macam Matriks 5. Matriks ortogonal Matriks A orthogonal jika dan hanya jika AT = A –1 = A-1 = B-1 (A-1)T = (AT)-1 A-1 AT Jika A adalah matriks orthogonal, maka(A-1)T = (AT)-1 Matriks

  16. 0 -1 1 0 0 1 -1 0 A = AT= ½√2 -½√2 ½√2 ½√2 ½√2 ½√2 -½√2 ½√2 B = BT= Kinds of Matrix 5. Orthogonal Matrix Matrix A is orthogonal if and only if AT = A –1 = A-1 = B-1 (A-1)T = (AT)-1 A-1 AT If A is orthogonal matrix, so(A-1)T = (AT)-1 Matriks

  17. 4 2 6 7 5 3 -9 7 A = Macam – macam Matriks Definisi: Transpose matriks A adalah matriks AT, kolom-kolomnya adalah baris-baris dari A, baris-barisnya adalah kolom-kolom dari A. 4 5 2 3 6 -9 77 AT = A’ = • [AT]ij = [A]ji n x m Jika A adalah matriks m x n, maka matriks transpose AT berukuran ……….. Matriks

  18. 4 2 6 7 5 3 -9 7 A = Kinds of Matrix Definisi: Transpose matrix A is matrix AT, its columns are rows of A, its rows is columns of A. 4 5 2 3 6 -9 77 AT = A’ = • [AT]ij = [A]ji n x m if A is matrix m x n, so matrix transpose AT should be ……….. Matriks

  19. 1 2 4 2 1 3 1 2 4 2 1 3 A = B = 1 2 2 2 1 3 2 1 2 2 1 3 C = D = 1 2 4 2 2 2 x 2 4 2 2 2 E = F = ? ? ? ? ? ? ? ? ? • 2 2 • 5 6 • 9 0 7 H = G = Macam – macam Matriks Kesamaan dua matriks • Dua matriks sama jika ukuran sama dan setiap entri yang bersesuaian sama. A = B C ≠ D E = F jika x = 1 2 2 2 5 4 6 G = H 9 0 7 Matriks

  20. 1 2 4 2 1 3 1 2 4 2 1 3 A = B = 1 2 2 2 1 3 2 1 2 2 1 3 C = D = 1 2 4 2 2 2 x 2 4 2 2 2 E = F = ? ? ? ? ? ? ? ? ? • 2 2 • 5 6 • 9 0 7 H = G = Kind of Matrix Similarity of two matrixes • Two matrix are similar if its size is similar and each symmetrical entry is similar A = B C ≠ D E = F jika x = 1 2 2 2 5 4 6 G = H 9 0 7 Matriks

  21. 4 2 2 3 4 2 2 3 A = A’ = Macam-macam Matriks Matriks Simetri Matriks A disebut simetris jika dan hanya jika A = AT A simetri 1 2 3 4 2 5 7 0 37 8 2 4 0 2 9 A = = AT Matriks

  22. 4 2 2 3 4 2 2 3 A = A’ = Kinds of Matrix Symmetrical matrix Matrix A is called symmetric if and only if A = AT A symmetric 1 2 3 4 2 5 7 0 37 8 2 4 0 2 9 A = = AT Matriks

  23. 4 2 6 7 5 3 -9 7 Macam-macam Matriks Sifat-sifat transpose matriks • Transpose dari A transpose adalah A: • (AT )T = A (AT)T A = A AT Contoh: 4 5 2 3 6 -9 77 4 5 2 3 6 -9 77 Matriks

  24. 4 2 6 7 5 3 -9 7 Kinds of Matrix properties of transpose matrix • Transpose of A transpose is A: • (AT )T = A (AT)T A = A AT Example: 4 5 2 3 6 -9 77 4 5 2 3 6 -9 77 Matriks

  25. T T T A A+B B = + • (A+B)T = • AT • BT + Macam-macam Matriks 2. (A+B)T = AT + BT Matriks

  26. T T T A A+B B = + • (A+B)T = • AT • BT + Kinds of Matrix 2. (A+B)T = AT + BT Matriks

  27. Macam-macam Matriks 3. (kA)T = k(A) T untuk skalar k T A kA T k • (kA)T = k(A)T Matriks

  28. Kinds of Matrix 3. (kA)T = k(A) T for scalar k T A kA T k • (kA)T = k(A)T Matriks

  29. T T T B AB A Macam-macam Matriks 4. (AB)T = BT AT = • (AB)T • = BTAT • AB Matriks

  30. T T T B AB A Kinds of Matrix 4. (AB)T = BT AT = • (AB)T • = BTAT • AB Matriks

  31. Macam-macam Matriks Soal : Isilah titik-titik di bawah ini • A simetri maka A + AT= …….. • ((AT)T)T = ……. • (ABC)T = ……. • ((k+a)A)T = …..... • (A + B + C)T = ………. • Kunci: • 2A • AT • CTBTAT • (k+a)AT • AT + BT + CT Matriks

  32. Kind of Matrix Quiz : Fill in the blanks bellow • A symmetric then A + AT= …….. • ((AT)T)T = ……. • (ABC)T = ……. • ((k+a)A)T = …..... • (A + B + C)T = ………. • Answer keys: • 2A • AT • CTBTAT • (k+a)AT • AT + BT + CT Matriks

  33. OPERASI MATRIKS Kompetesi Dasar Menyelesaikan Operasi Matriks Indikator • Dua matriks atau lebih ditentukan hasil penjumlahan atau pengurangannya • Dua matriks atau lebih ditentukan hasil kalinya Matriks

  34. OPERATION OF MATRIX Basic competence Finishing operation matrix Indicator • Two or more matrixes is defined by the result of their addition or subtraction • Two or more matrixes is defined by the result of their multiplication Matriks

  35. 10 22 1 -1 2 6 7 5 A = B = 10+2 22+6 1+7 -1+5 12 28 8 4 A + B = = 10-2 22-6 1-7 -1-5 8 16 -6 -6 A - B = = OPERASI MATRIKS Penjumlahan dan pengurangan dua matriks Contoh : Matriks

  36. 10 22 1 -1 2 6 7 5 A = B = 10+2 22+6 1+7 -1+5 12 28 8 4 A + B = = 10-2 22-6 1-7 -1-5 8 16 -6 -6 A - B = = OPERATION OF MATRIX Addition and subtraction of two matixes Example: Matriks

  37. OPERASI MATRIKS Apa syarat agar dua matriks dapat dijumlahkan? Jawab: Ordo dua matriks tersebut sama • A = [aij] dan B = [bij] berukuran sama, • A + B didefinisikan: (A + B)ij = (A)ij + (B)ij = aij + bij Matriks

  38. OPERATION OF MATRIX What is the condition so that two matrixes can be added? Answer: The ordo of the two matrixes are the same • A = [aij] dan B = [bij] have the same size, • A + B is defined: (A + B)ij = (A)ij + (B)ij = aij + bij Matriks

  39. 1 4 -9 3 7 0 5 9 -13 7 3 1 -2 4 -5 9 -4 3 K = L = 25 30 5 35 10 15 5 6 1 7 2 3 C = D = ? ? ? ? ? ? C + D = ? ? ? ? ? ? ? ? ? K + L = OPERASI MATRIKS Jumlah dua matriks D + C = L + K = Apa kesimpulanmu? Apakah jumlahan matriks bersifat komutatif? Matriks

  40. 1 4 -9 3 7 0 5 9 -13 7 3 1 -2 4 -5 9 -4 3 K = L = 25 30 5 35 10 15 5 6 1 7 2 3 C = D = ? ? ? ? ? ? C + D = ? ? ? ? ? ? ? ? ? K + L = OPERATION OF MATRIX The quantity of two matrixes D + C = L + K = What is your conclusion? Is the addition of matrixes commutative? Matriks

  41. -8 0 • 4 7 2 • -1 8 4 • 6 -1 2 • 9 9 8 • -2 16 8 • 7 2 • 5 2 6 • -1 8 4 D = • 7 2 • 5 2 6 C = C +D = E = 0 0 0 0 0 0 0 0 0 0 0 0 A = B = OPERASI MATRIKS • Soal: • C + D =… • C + E = … • A + B = … Feedback: Matriks

  42. -8 0 • 4 7 2 • -1 8 4 • 6 -1 2 • 9 9 8 • -2 16 8 • 7 2 • 5 2 6 • -1 8 4 D = • 7 2 • 5 2 6 C = C +D = E = 0 0 0 0 0 0 0 0 0 0 0 0 A = B = OPERATION OF MATRIX • Exercise: • C + D =… • C + E = … • A + B = … Feedback: Matriks

  43. 5 6 1 7 2 3 A = OPERASI MATRIKS Hasil kali skalar dengan matriks 5x5 25 5x1 5 5x6 30 5A = = 5x5 35 5x3 15 5x2 10 Apa hubungan H dengan A? 250 300 50 350 100 150 H = 50 H = A • Diberikan matriks A = [aij] dan skalar c, perkalian skalar cA mempunyai entri-entri sebagai berikut: • (cA)ij = c.(A)ij = caij • Catatan: Pada himpunan Mmxn, perkalian matriks dengan skalar bersifat tertutup (menghasilkan matriks dengan ordo yang sama) Matriks

  44. 5 6 1 7 2 3 A = OPERATION OF MATRIX The multiplication result of scalar matrix 5x5 25 5x1 5 5x6 30 5A = = 5x5 35 5x3 15 5x2 10 What is the relation between H and A? 250 300 50 350 100 150 H = 50 H = A • Given matrix A = [aij] aand scalar c, the multiplication of scalar cA have the following entries: • (cA)ij = c.(A)ij = caij • Note: In the set of Mmxn, the matrix multiplication with scalar have closed properties (it will have matrix with the same orrdo) Matriks

  45. 1 4 -9 3 7 0 5 9 -13 K = 4 16 -36 12 28 0 20 36 -52 4K = 5 20 -45 15 35 0 25 45 -65 5K = OPERASI MATRIKS • K 3 x 3 Matriks

  46. 1 4 -9 3 7 0 5 9 -13 K = 4 16 -36 12 28 0 20 36 -52 4K = 5 20 -45 15 35 0 25 45 -65 5K = OPERATION OF MATRIX • K 3 x 3 Matriks

  47. 7 2 • 5 2 6 A = 0 0 0 0 0 0 0 0 0 0 0 0 A = = 0*2 0*7 0*2 0*5 0*2 0*6 7*0 7*0 7*0 7*0 7*0 7*0 cA = cA = OPERASI MATRIKS • Diketahui bahwa cA adalah matriks nol. Apa kesimpulan Anda tentang A dan c? Contoh: c = 7 c = 0 kesimpulan Kasus 1: c = 0 dan A matriks sembarang. Kasus 2: A matriks nol dan c bisa berapa saja. Matriks

  48. 7 2 • 5 2 6 A = 0 0 0 0 0 0 0 0 0 0 0 0 A = = 0*2 0*7 0*2 0*5 0*2 0*6 7*0 7*0 7*0 7*0 7*0 7*0 cA = cA = OPERATION OF MATRIX • Known that cA is zero matrix. What is your conclusion about A and c? Example: c = 7 c = 0 Conclusion Case 1: c = 0 and A is any matrix Case 2: A is zero matrix and c can be any number Matriks

  49. ∑ aikbkj = ai1b1j +ai2b2j+………airbrj k = 1 1 2 7 -6 4 -9 2 3 4 5 8 -7 9 -4 1 -5 7 -8 A = B = OPERASI MATRIKS Perkalian matriks dengan matriks • Definisi: • Jika A = [aij] berukuran m x r , dan B = [bij] berukuran r x n, maka matriks hasil kali A dan B, yaitu C = AB mempunyai elemen-elemen yang didefinisikan sebagai berikut: r • (C)ij = (AB)ij = A B AB • Syarat: r xn m xn m xr Tentukan AB dan BA Matriks

  50. ∑ aikbkj = ai1b1j +ai2b2j+………airbrj k = 1 1 2 7 -6 4 -9 2 3 4 5 8 -7 9 -4 1 -5 7 -8 A = B = OPERATION OF MATRIX Multiplication between matrix • Definition: • If A = [aij] have size m x r , and B = [bij] have size r x n, then the matrix which is from the multiplication result between A and B, yaitu is C = AB has elements that defined as follows: r • (C)ij = (AB)ij = A B AB • Condition: r xn m xn m xr Define AB and BA Matriks

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