MATRIX Concept of Matrix Matrik

Kinds of Matrix Basic Competences : Indicators : Describing the kinds of matrix Indicators : Matrix is determined by its elements and notations Matriks matrix is distinguished by its kinds and relations Hal.: 2 Matriks Matrik

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 A = rows Notation: Matrix: A = [aij] Element: (A)ij = aij Order A: m x n column Hal.: 3 Matriks Matrik

Kinds of Matrix 1. Row matrix Row matrix is a matrix which consists of one row. 2 5 1 -8 25 -2 0 14 8 Hal.: 4 Matriks Matrik

Kinds of Matrix 2. Column matrix Column matrix is a matrix which consists of one column. 2 -7 9 2 1 Hal.: 5 Matriks Matrik

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. Hal.: 6 Matriks Matrik

4. Zero matrix Kinds of Matrix zero matrix is a matrix which all of its elements are zero. 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 Hal.: 7 Matriks Matrik

5. Orthogonal Matrix (A-1)T = (AT)-1 A-1 AT Kinds of Matrix 5. Orthogonal Matrix Matrix A is orthogonal if and only if AT = A –1 0 -1 1 0 A = 0 1 -1 0 AT= = A-1 B = ½√2 -½√2 ½√2 ½√2 BT= ½√2 ½√2 -½√2 ½√2 = B-1 Matriks ortogonal adalah matriks yang inversenya sama dengan transposenya. A-1 AT (A-1)T = (AT)-1 If A is orthogonal matrix, so (A-1)T = (AT)-1 Hal.: 8 Matriks Matrik

Kinds of Matrix [AT]ij = [A]ji Definisi: 4 5 2 3 6 -9 7 7 4 2 6 7 5 3 -9 7 A = AT = A’ = Definisi: Transpose matrix A is matrix AT, its columns are rows of A, its rows is columns of A. Transpose matriks Transpose dari matriks adalah matriks baru yang kolom –kolom menjadi baris-baris.  Matriks simetri adalah matriks yang sama dengan transposenya. Matriks orthogonal adalah matriks yang inversenya sama dengan trnsposenya. Transpose dari transpose A adalaha A. Jumlahan dua matriks sama dengan jumlahan transposenya, transpose dari skalar kali matriks sama dengan skalar kali trenasposenya. Transpose dari hasil kali sama dengan hasil kali transposenya dengan urutan terbalik. Tampilkan contoh matriks 2x4, kmd transposekan  Berikan definisi umum dari transpose.  Matriks A simetri jkk A = AT, berikan contoh  Matriks A orthogonal jika dan hanya jika AT = A –1, berikan contoh: matriks rotasi.  Berikan contoh dan rumus umum (A-1)T = (AT)-1 [AT]ij = [A]ji n x m if A is matrix m x n, so matrix transpose AT should be ……….. Hal.: 9 Matriks Matrik

Similarity of two matrixes Kind of Matrix Similarity of two matrixes Two matrix are similar if its size is similar and each symmetrical entry is similar 1 2 4 2 1 3 A = 1 2 4 2 1 3 B = A = B 1 2 2 2 1 3 C = 2 1 2 2 1 3 D = C ≠ D 1 2 4 2 2 2 E = x 2 4 2 2 2 F = E = F jika x = 1 2 2 2 H = ? ? ? 2 2 5 6 9 0 7 G = 4 5 6 G = H 9 7 Hal.: 10 Matriks Matrik

Symmetrical matrix Matrix A is called symmetric if and only if A = AT Kinds of Matrix Symmetrical matrix Matrix A is called symmetric if and only if A = AT 4 2 2 3 A = 4 2 2 3 A’ = A symmetric Matriks simetri adalah matriks yang sama dengan transposenya. Matriks ortogonal adalah matriks yang inversenya sama dengan transposenya. 1 2 3 4 2 5 7 0 3 7 8 2 4 0 2 9 A = = AT Hal.: 11 Matriks Matrik

properties of transpose matrix Kinds of Matrix properties of transpose matrix Transpose of A transpose is A: (AT )T = A A (AT)T = A AT Example: Transpose dari transpose adalah matriks iu sendiri 4 5 2 3 6 -9 7 7 4 5 2 3 6 -9 7 7 4 2 6 7 5 3 -9 7 Hal.: 12 Matriks Matrik

= (A+B)T AT BT Kinds of Matrix 2. (A+B)T = AT + BT A B A+B T T T + 2.Transpose dari jumlaham nmatriks sama dengan jumlahan transpose-transposenya Hal.: 13 Matriks Matrik

(kA)T = k(A)T Kinds of Matrix 3. (kA)T = k(A) T for scalar k k T T kA 3. Transpose dari hasil kali skalr dengan matriks sama dengan hasil kali skalar dengan transposenya (kA)T = k(A)T Hal.: 14 Matriks Matrik

= (AB)T AB = BTAT Kinds of Matrix 4. (AB)T = BT AT B T T A AB T 4. Transpose hasil kali A dan B sama dengan hasil kali transpose a dan transpose B. (AB)T AB = BTAT Hal.: 15 Matriks Matrik

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 QUIZMAKER Hal.: 16 Matriks Matrik

Basic competence OPERATION OF MATRIX 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 Hal.: 17 Matriks Matrik

Addition and subtraction of two matixes OPERATION OF MATRIX Addition and subtraction of two matixes Example: 10 22 1 -1 A = 2 6 7 5 B = 10+2 22+6 1+7 -1+5 A + B = 12 28 8 4 = A - B = 10-2 22-6 1-7 -1-5 8 16 -6 -6 = Hal.: 18 Matriks Matrik

What is the condition so that two matrixes can be added? 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 Hal.: 19 Matriks Matrik

The quantity of two matrixes OPERATION OF MATRIX The quantity of two matrixes 1 4 -9 3 7 0 5 9 -13 K = 7 3 1 -2 4 -5 9 -4 3 L = 25 30 5 35 10 15 D = 5 6 1 7 2 3 C = C + D = ? ? ? D + C = K + L = ? ? ? L + K = What is your conclusion? Is the addition of matrixes commutative? Hal.: 20 Matriks Matrik

OPERATION OF MATRIX Exercise: Feedback: -8 0 4 7 2 -1 8 4 C = D = 7 2 C + E = … A + B = … -8 0 4 7 2 -1 8 4 C = D = 7 2 5 2 6 -1 8 4 E = 7 2 5 2 6 0 0 0 A = 0 0 0 B = 6 -1 2 9 9 8 -2 16 8 C +D = Feedback: Hal.: 21 Matriks Matrik

The multiplication result of scalar matrix OPERATION OF MATRIX The multiplication result of scalar matrix 5x5 5x6 5x1 25 30 5 5 6 1 7 2 3 A = 5A = = 5x5 5x2 5x3 35 10 15 What is the relation between H and A? 250 300 50 350 100 150 H = H = 50 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) Hal.: 22 Matriks Matrik

OPERATION OF MATRIX 4K = 5K = K 3 x 3 1 4 -9 3 7 0 K = 5 9 -13 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 = Hal.: 23 Matriks Matrik

OPERATION OF MATRIX Known that cA is zero matrix. What is your conclusion about A and c? Example: 0 0 0 A = A = 7 2 5 2 6 c = 7 c = 0 cA = 0*2 0*7 0*2 0*5 0*2 0*6 cA = 7*0 7*0 7*0 0 0 0 = Conclusion Case 1: c = 0 and A is any matrix Case 2: A is zero matrix and c can be any number Hal.: 24 Matriks Matrik

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 ∑ aikbkj = ai1b1j +ai2b2j+………airbrj k = 1 (C)ij = (AB)ij = A B AB Condition: m x r r x n m x n 1 2 7 -6 4 -9 B = 2 3 4 5 8 -7 9 -4 1 -5 7 -8 A = Define AB and BA Hal.: 25 Matriks Matrik

The multiplication between matrixes OPERATION OF MATRIX The multiplication between matrixes Example: 1 2 7 -6 4 -9 11 3 B = 2 3 4 5 8 -7 9 -4 1 -5 7 -8 A = = 2.1 +3.7+4.4+5.11 -35 -49 -35 -94 -55 94 -35 -49 -35 -94 -55 A B = = BA is not define Hal.: 26 Matriks Matrik

OPERATION OF MATRIX ABmxm ABnxn n x k m x n n x k m x n 1. Given A and B, AB and BA is defined. What is your conclusion? A B n x k m x n B A n x k m x n m = k AB and BA square matricx ABmxm ABnxn 2. AB = O is zero matrix, is one of (A or B) is zero matrix? 2 3 A = 3 -3 -2 2 B = AB = 0 0 AB is zero matrix. Matrix A and B is not certain zero matrix Hal.: 27 Matriks Matrik

OPERATION OF MATRIX Example 1: Define the multiplication result if it defined: A B = ?? AC = ?? BD = ?? CD = ?? DB = ?? 2 3 4 5 4 7 9 0 2 3 5 6 A = 1 2 -9 0 8 0 5 6 B = 7 -11 4 3 5 -6 C = 1 8 9 5 6 2 5 6 -9 0 0 -4 7 8 9 D = Hal.: 28 Matriks Matrik

OPERATION OF MATRIX Example 2: A0 = I An = A A A …A n factor 2 3 1 2 A = 2 3 1 2 2 3 1 2 A2 = 2 3 1 2 2 3 1 2 2 3 1 2 A3 = A x A2 = A0 = I An = n factor An+m = An Am A A A …A Hal.: 29 Matriks Matrik

DETERMINANT AND INVERSE Basic Competence: Define the determinant and inverse Indicator : Matrix is defined by its determinant Matrix is defined by its inverse Hal.: 30 Matriks Matrik

DETERMINANT AND INVERSE Determinant Matrix ordo 2 x 2 Determinant value of a matrix ordo 2 x 2 is the multiplication result of the main diagonal elements and subtract by the multiplication result of the second diagonal. For example, known matrix A ordo 2 x 2, A = Determinant A is det A = = ad - bc Hal.: 31 Matriks Matrik

Example: Matrix inverse 2x2 DETERMINANT AND INVERSE Example: Matrix inverse 2x2 3 2 4 1 A = Berikut contoh penerapan rumus menentukan inverse matriks 2x2 A-1 = = I Hal.: 32 Matriks Matrik

DETERMINANT AND INVERSE Example : 1. When matrix Doesn’t have inverse? ad-bc = 0 2. Define the following matrix inverse 5 1 1 2 a. 2/3 -1/5 -1/5 5/3 a. 0 1 0 2 b. b. Doesn’t have inverse QUIZMAKER 0 0 4 1 c. c. Doesn’t have inverse 1 0 0 1 d. 1 0 0 1 d. Hal.: 33 Matriks Matrik

DETERMINANT AND INVERSE B is inverse of matrix A, if AB = BA = I matrix identities, it is written B = A-1 A A-1 A-1 A I = = If A = , then Diberikan matriks persegi A, matriks manakah jika dikalikan dengan A hasilnya matriks identitas? Jawabnya adalah inverse dari A. Marilah kita definisikan inverse matriks. B adalah inverse dari matriks A, jika AB = BA = I matriks identitas, ditulis B = A inverse (seperti pangkat -1). Perhatikan contoh berikut: A kali A inverse sama dengan A inverse kali A sama dengan matriks identitias I. B kali B inverse sama dengan B inverse kali B sama dengan matriks identitas I. Masalah berikutnya adalah: Apakah setiap matriks mempunyai inverse? Jika mempunyai, bagaimana menentukannya? Apakah inverse matriks (jika ada0 adalah tungga? Hal.: 34 Matriks Matrik

DETERMINANT AND INVERSE Example 1 : Defined the inverse of matrix Answer : det B = (-5) . (-4) – (-2) . (-10) = 20 – 20 = 0 , So, matrix B doesn’t have inverse Hal.: 35 Matriks Matrik

DETERMINANT AND INVERSE Example 2 : Known matrix Show that A.A-1 = A-1.A = I and B.B-1 = B-1. B = I 4 2 2 2 ½ -½ -½ 1 ½ -½ -½ 1 4 2 2 2 1 0 0 1 = = A A-1 A-1 A I Diberikan matriks persegi A, matriks manakah jika dikalikan dengan A hasilnya matriks identitas? Jawabnya adalah inverse dari A. Marilah kita definisikan inverse matriks. B adalah inverse dari matriks A, jika AB = BA = I matriks identitas, ditulis B = A inverse (seperti pangkat -1). Perhatikan contoh berikut: A kali A inverse sama dengan A inverse kali A sama dengan matriks identitias I. B kali B inverse sama dengan B inverse kali B sama dengan matriks identitas I. Masalah berikutnya adalah: Apakah setiap matriks mempunyai inverse? Jika mempunyai, bagaimana menentukannya? Apakah inverse matriks (jika ada0 adalah tungga? 4 2 1 2 2 1 3 3 1 ½ -½ 1 -½ -½ 1 0 3 -2 4 2 1 2 2 1 3 3 1 ½ -½ 1 -½ -½ 1 0 3 -2 1 0 0 0 1 0 0 0 1 = = B B-1 B-1 B I Hal.: 36 Matriks Matrik

DETERMINANT AND INVERSE Matrix ordo 3 x 3 Matrix Determinant Ordo 3 x 3 With Sarrus rule, determinant A is as follows _ _ _ + + + Hal.: 37 Matriks Matrik

DETERMINANT AND INVERSE The equation of linear with two variable using matrix For example SPL The equation can be changed into the following matrix Hal.: 38 Matriks Matrik

DETERMINANT AND INVERSE Then can be write as Example Hal.: 39 Matriks Matrik

DETERMINANT AND INVERSE Example: Define the value of x and y that fulfill the equation of linear system answer : Equation system If in matrix Hal.: 40 Matriks Matrik

DETERMINANT AND INVERSE The matrix multiplication in the form of AP =B with So, the value of x = 5 and y = 2 Hal.: 41 Matriks Matrik

DETERMINANT AND INVERSE The solution of linear equation system with two variables using determinant or Cramer rule For example SPL Then, with Cramer rule, we get dan Hal.: 42 Matriks Matrik

DETERMINANT AND INVERSE Example : Use the Cramer rule to define the solution set of linear equation system answer : With cramer rule, we get So, the solution set is {(1,2)}. Hal.: 43 Matriks Matrik

DETERMINANT AND INVERSE Finishing the equation of linear system with three variables using matrix SPL in the form of: It can be written in the form of matrix equation: a11x1 + a12x2 + a13x3 +….. ..a1nxn = b1 a21x1 + a22x2 + a23x3 +…….a2nxn = b2 am1x1 + am2x2 + am3x3 + ……amnxn = bm a11 a12……...a1n a21 a22 ……..a2n : : : am1 am2…… amn = x1 x2 : xn b1 b2 : bn x b A: matrix coefficient Ax = b Hal.: 44 Matriks Matrik

DETERMINANT AND INVERSE Example : x1 + 2x2 + x3 = 6 -x2 + x3 = 1 4x1 + 2x2 + x3 = 4 SPL It can be written in the form of the following matrix 1.x1 +2.x2 + 1.x3 0.x1 + -1.x2 + 1.x3 4.x1 +2.x2 + 1.x3 6 1 4 1 2 1 0 -1 1 4 2 1 x1 x2 x3 6 1 4 = = Hal.: 45 Matriks Matrik

The multiplication of identity matrix DETERMINANT AND INVERSE The multiplication of identity matrix A= 1 2 3 7 5 6 -9 3 -7 1 2 3 7 5 6 -9 3 -7 1 0 0 0 1 0 0 0 1 1 2 3 7 5 6 -9 3 -7 A.I = X = Masih ingat hasil kali matriks dengan matriks identitas? Marilah kita kalikan A dengan matriks identitas I, baik dari kiri maupun dari kanan. Bagaimana hasilnya? Apa kesimpulanmu? Perkalian matriks persegi dengan matriks identitas berukuran sama bersifat komutatif. Hasil kalinya sama dengan matriks A. 1 2 3 7 5 6 -9 3 -7 1 2 3 7 5 6 -9 3 -7 1 0 0 0 1 0 0 0 1 X = I.A = Hal.: 46 Matriks Matrik

DETERMINANT AND INVERSE AB = A and BA = A, what is your conclusion? 1 4 -9 3 7 0 5 9 -13 1 0 0 0 1 0 0 0 1 1 4 -9 3 7 0 5 9 -13 = 1 0 0 0 1 0 0 0 1 1 4 -9 3 7 0 5 9 -13 1 4 -9 3 7 0 5 9 -13 = QUIZMAKER Feedback: A dan B matriks persegi dengan ordo sama B adalah matriks identitas A I I A A = = AB = A and BA = A, then B = I (I identity matrix ) Hal.: 47 Matriks Matrik

DETERMINANT AND INVERSE 4 2 2 2 ½ -½ -½ 1 1 0 0 1 = A A-1 I a b c d A-1 1 0 0 1 = Marilah kita menentukan inverse matriks 2x2. Perhatikan bahwa A kali A inverse adalah I. Jika A diketahui, bagaimanan menentukan A inverse? A-1 d -b -c a 1 ad - bc = = If ad –bc = 0 then A doesn’t have inverse Hal.: 48 Matriks Matrik

DETERMINANT AND INVERSE If there is inverse of matrix is only one: If B = A-1 and C = A-1, then B = C (A-1)-1 2. (A-1)-1 = A 4 2 2 2 A = ½ -½ -½ 1 ? 1 0 0 1 = A-1 = ½ -½ -½ 1 A-1 Sifat-sifat matriks inverse Jika A mempunyai inverse, maka inversenya tunggal. Inverse dari inverse matriks A adalah A sendiri. Matriks inverse dari matriks pangkat n sama dengan inverse matriks dipangkatkan n. 1-5 dalam bentuk rumus, kemudian contoh-contoh sederhana Quiz: menentukan inverse matriks 2x2, inverse dari transposenya, berikan 2 matriks, satu rtogonal satu tidak, identifikasi. Untuk matriks ort, hitung determinan.  Link ke bukti sifat 1, 2 4 2 2 2 A Hal.: 49 Matriks Matrik

DETERMINANT AND INVERSE If A have inverse then An have inverse and (An)-1 = (A-1)n, n = 0, 1, 2, 3,… 4 2 2 2 A = ½ -½ -½ 1 A-1 = 4 2 2 2 4 2 2 2 4 2 2 2 104 64 64 40 A3 = = (A3)-1 = 0.625 -1 -1 1.625 Jika A adalah matriks persegi yang mempunyai inverse, maka menghitung inverse kemudian memangatkan hasilnya sama dengan memangkatkan dahulu kemudian dihitung inversenya. The same with (A-1)3 = ½ -½ -½ 1 ½ -½ -½ 1 ½ -½ -½ 1 0.625 -1 -1 1.625 = Hal.: 50 Matriks Matrik

DETERMINANT AND INVERSE 4. (AB)-1 = B-1 A-1 3 5 2 2 B = B-1 = ½ 5/4 ½ - ¾ 4 2 2 2 A = 16 24 10 14 -0.875 1.5 0.625 -1 (AB)-1 = -1 = Jika A dan B dapat dikalikan dan masing-masing mempunyai inverse, maka AB juga mempunyai inverse. Inverse AB sama dengan hasil kali B inverse deang A inverse (urutan terbalik) ½ 5/4 ½ - ¾ ½ -½ -½ 1 -0.875 1.5 0.625 -1 B-1 A-1 = = ½ -½ -½ 1 ½ 5/4 ½ - ¾ -0.5 1 0.75 -1.375 A-1 B-1 = = Hal.: 51 Matriks Matrik