Konsep Matriks

Matriks

Concept of Matrix

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Macam-macam Matriks

Kompetensi Dasar :Mendeskripsikan macam-macam matriks

Indikator :Matriks ditentukan unsur dan notasinyaMatriks dibedakan menurut jenis dan relasinya

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Kinds of Matrix

Basic Competences :Describing the kinds of matrix

Indicators :Matrix is determined by its elements and notationsMatriks matrix is distinguished by its kinds and relations

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Pengertian MatriksMatriks adalah susunan bilangan-bilangan yang terdiri atas baris-baris dan kolom-kolom.

a11 a12.a1j a1na21 a22 a2j.a2n : : : :ai1 ai2 aij.. ain: : : :am1 am2amj. amn

A = bariskolomNotasi: Matriks: A = [aij]Elemen: (A)ij = aijOrdo A: m x nMasing-masing bilangan dalam matriks disebut entri atau elemen. Ordo (ukuran) matriks adalah jumlah baris kali jumlah kolom. Macam macam Matriks

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Definition of MatrixMatrix is the arrangement of numbers which consists of rows and columns.

a11 a12.a1j a1na21 a22 a2j.a2n : : : :ai1 ai2 aij.. ain: : : :am1 am2amj. amn

A = rowscolumnNotation: Matrix: A = [aij]Element: (A)ij = aijOrder A: m x nEach 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. Kinds of Matrix

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Macam-macam MatriksMatriks baris adalah matriks yang hanya terdiri dari satu baris.

1. Matriks Baris

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Kinds of MatrixRow matrix is a matrix which consists of one row.

1. Row matrix

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Macam-macam Matriks2. Matriks KolomMatriks Kolom adalah matriks yang hanya terdiri dari satu kolom

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Kinds of Matrix2. Column matrixColumn matrix is a matrix which consists of one column.

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3. Matriks PersegiMatriks persegi (bujur sangkar) adalah matriks yang jumlah baris dan jumlah kolom sama.1 2 42 2 23 3 3Trace(A) = 1 + 2 + 3Trace dari matriks adalah jumlahan elemen-elemen diagonal utamadiagonal utama Macam macam Matriks

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3. Square matrixSquare matrix is a matrix which has the same numbers of rows and columns.1 2 42 2 23 3 3Trace(A) = 1 + 2 + 3Trace from matrix is the total numbers from the main diagonal elements.Main diagonal Kinds of Matrix

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4. Matriks Nol Matriks nol adalah matriks yang semua elemennya nol

0 0 00 00 01 00 11 0 00 1 00 0 11 0 0 00 1 0 00 0 1 0 0 0 0 1I2I3I4Matriks identitas adalah matriks persegi yang elemen diagonal utamanya 1 dan elemen lainnya 0 Macam- macam Matriks

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4. Zero matrix zero matrix is a matrix which all of its elements are zero.

0 0 00 00 01 00 11 0 00 1 00 0 11 0 0 00 1 0 00 0 1 0 0 0 0 1I2I3I4Matrix identity is a square matrix which its main diagonal element is 1 and the other element is 0. Kinds of Matrix

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5. Matriks ortogonalMatriks A orthogonal jika dan hanya jika AT = A 1 Jika A adalah matriks orthogonal, maka (A-1)T = (AT)-1= A-1= B-1(A-1)T = (AT)-1 A-1 AT Macam-macam Matriks

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5. Orthogonal MatrixMatrix A is orthogonal if and only if AT = A 1 If A is orthogonal matrix, so (A-1)T = (AT)-1= A-1= B-1(A-1)T = (AT)-1 A-1 AT Kinds of Matrix

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Macam macam MatriksDefinisi:Transpose matriks A adalah matriks AT, kolom-kolomnya adalah baris-baris dari A, baris-barisnya adalah kolom-kolom dari A.

AT = A = 4 5 2 36 -9 7 7Jika A adalah matriks m x n, maka matriks transpose AT berukuran ..[AT]ij = [A]ji n x m

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Kinds of MatrixDefinisi:Transpose matrix A is matrix AT, its columns are rows of A, its rows is columns of A.

AT = A = 4 5 2 36 -9 7 7if A is matrix m x n, so matrix transpose AT should be ..[AT]ij = [A]ji n x m

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Kesamaan dua matriksDua matriks sama jika ukuran sama dan setiap entri yang bersesuaian sama.A = BC DE = F jika x = 1G = H222456907 Macam macam Matriks

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Similarity of two matrixesTwo matrix are similar if its size is similar and each symmetrical entry is similarA = BC DE = F jika x = 1G = H222456907 Kind of Matrix

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Matriks SimetriMatriks A disebut simetris jika dan hanya jika A = ATA simetri1 2 3 42 5 7 0 3 7 8 2 4 0 2 9A == AT Macam-macam Matriks

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Symmetrical matrixMatrix A is called symmetric if and only if A = ATA symmetric1 2 3 42 5 7 0 3 7 8 2 4 0 2 9A == AT Kinds of Matrix

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Sifat-sifat transpose matriksAAT(AT)T(AT )T = ATranspose dari A transpose adalah A:4 5 2 36 -9 7 74 5 2 36 -9 7 7= AContoh: Macam-macam Matriks

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properties of transpose matrixAAT(AT)T(AT )T = ATranspose of A transpose is A:4 5 2 36 -9 7 74 5 2 36 -9 7 7= AExample: Kinds of Matrix

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Macam-macam Matriks2. (A+B)T = AT + BT

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Kinds of Matrix2. (A+B)T = AT + BT

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Macam-macam Matriks3. (kA)T = k(A) T untuk skalar k

kA(kA)T = k(A)TATTk

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Kinds of Matrix3. (kA)T = k(A) T for scalar k

kA(kA)T = k(A)TATTk

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Macam-macam Matriks4. (AB)T = BT AT (AB)T =AB= BTAT

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Kinds of Matrix4. (AB)T = BT AT (AB)T =AB= BTAT

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Macam-macam MatriksIsilah titik-titik di bawah iniA simetri maka A + AT= ..((AT)T)T = .(ABC)T = .((k+a)A)T = .....(A + B + C)T = .Kunci:2A ATCTBTAT (k+a)AT AT + BT + CT

Soal :

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Kind of MatrixFill in the blanks bellowA symmetric then A + AT= ..((AT)T)T = .(ABC)T = .((k+a)A)T = .....(A + B + C)T = .Answer keys:2A ATCTBTAT (k+a)AT AT + BT + CT

Quiz :

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OPERASI MATRIKSKompetesi DasarMenyelesaikan Operasi MatriksIndikatorDua matriks atau lebih ditentukan hasil penjumlahan atau pengurangannyaDua matriks atau lebih ditentukan hasil kalinya

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OPERATION OF MATRIXBasic competenceFinishing operation matrixIndicatorTwo 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

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Contoh :

Penjumlahan dan pengurangan dua matriks OPERASI MATRIKS

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Example:

Addition and subtraction of two matixes OPERATION OF MATRIX

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OPERASI MATRIKSApa 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

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OPERATION OF MATRIXWhat is the condition so that two matrixes can be added?

Answer:The ordo of the two matrixes are the sameA = [aij] dan B = [bij] have the same size,

A + B is defined: (A + B)ij = (A)ij + (B)ij = aij + bij

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Jumlah dua matriks

D + C = L + K = Apa kesimpulanmu? Apakah jumlahan matriks bersifat komutatif? OPERASI MATRIKS

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The quantity of two matrixes

D + C = L + K = What is your conclusion? Is the addition of matrixes commutative? OPERATION OF MATRIX

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OPERASI MATRIKSSoal:

C + D =C + E = A + B =

Feedback:

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OPERATION OF MATRIXExercise:

C + D =C + E = A + B =

Feedback:

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Hasil kali skalar dengan matriks5A = =250 300 50350 100 150H = H =Diberikan matriks A = [aij] dan skalar c, perkalian skalar cA mempunyai entri-entri sebagai berikut:(cA)ij = c.(A)ij = caijApa hubungan H dengan A?5x55x55x65x25x15x325353010515Catatan: Pada himpunan Mmxn, perkalian matriks dengan skalar bersifat tertutup (menghasilkan matriks dengan ordo yang sama)50A OPERASI MATRIKS

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The multiplication result of scalar matrix5A = =250 300 50350 100 150H = H =Given matrix A = [aij] aand scalar c, the multiplication of scalar cA have the following entries:(cA)ij = c.(A)ij = caijWhat is the relation between H and A?5x55x55x65x25x15x325353010515Note: In the set of Mmxn, the matrix multiplication with scalar have closed properties (it will have matrix with the same orrdo)50A OPERATION OF MATRIX

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OPERASI MATRIKSK 3 x 3

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OPERATION OF MATRIXK 3 x 3

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OPERASI MATRIKSDiketahui bahwa cA adalah matriks nol. Apa kesimpulan Anda tentang A dan c?

c = 0c = 7Kasus 1: c = 0 dan A matriks sembarang. Kasus 2: A matriks nol dan c bisa berapa saja.Contoh:kesimpulan

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OPERATION OF MATRIXKnown that cA is zero matrix. What is your conclusion about A and c?

c = 0c = 7Case 1: c = 0 and A is any matrixCase 2: A is zero matrix and c can be any numberExample:Conclusion

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OPERASI MATRIKSDefinisi:J