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Matrices in Discrete Mathematics and its Applications

Definition Of Matrix
Square Matrix with Example
Matrix Operations
Matrix Multiplication with Example
Transposes and Power Matrices
Transposes of a Matrix with Example
Symmetric Matrix with Example
Zero-One Matrices
Properties of Matrix

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Matrices in Discrete Mathematics and its Applications

  1. 1. Discrete Mathematics And Its Applications Chapter 2 Notes 2.6 Matrices Lecture Slides By Adil Aslam mailto:adilaslam5959@gmail.com
  2. 2. Definition Of Matrix • A matrix is a rectangular array of numbers. A matrix with m rows and n columns is called an m x n matrix. The plural of matrix is matrices. A matrix with the same number of rows as columns is called square. Two matrices are equal if they have the same number of rows and the same number of columns and the corresponding entries in every position are equal. • Example: The matrix is a 3 X 2 matrix. 31 20 11           Lecture Slides By Adil Aslam 2
  3. 3. Definition 2 • Let • The ith row of A is the 1 x n matrix [ai1, ai2, …, ain]. The jth column of A is the n x 1 matrix . ... 2n1 a 2 ... 2221 1 ... 1211                nn a n a n aaa n aaa A  anj 2 1                j j a a Lecture Slides By Adil Aslam 3
  4. 4. Definition 2 cont. • The (i, j)th element or entry of A is the element aij, that is, the number in the ith row and jth column of A. • A convenient shorthand notation for expressing the matrix A is to write A = [aij], which indicates that A is the matrix with its (i, j)th element equal to aij. Lecture Slides By Adil Aslam 4
  5. 5. Matrix • Element • Each value in a matrix; either a number or a constant. • Dimension • Number of rows by number of columns of a matrix. • A matrix is named by its dimensions. • Examples: Find the dimensions of each matrix Dimensions: 3x2 Dimensions: 2x4 Dimensions: 4x1 Lecture Slides By Adil Aslam 5
  6. 6. Matrix • Row Matrix • A matrix with only one row. • [1 x n] matrix • Column Matrix • A matrix with only one column. • [m x 1] matrix    jn aaaaA ,,21  Lecture Slides By Adil Aslam 6
  7. 7. Square Matrix A square matrix is a matrix that has the same number of rows and columns (n  n) • Example B  5 4 7 3 6 1 2 1 3         Lecture Slides By Adil Aslam 7
  8. 8. Equal matrices • Two matrices A = [aij] and B = [bij] are said to be equal (A = B) iff each element of A is equal to the corresponding element of B, i.e., aij = bij for 1  i  m, 1  j  n. Lecture Slides By Adil Aslam 8 Given that A = B, find a, b, c and d. 1 0 4 2 A       a b B c d        Example: and if A = B, then a = 1, b = 0, c = -4 and d = 2.
  9. 9. Matrix Operations • Matrix Equality • Let A and B be two matrices. A=B if they have the same number of rows and columns, and every element at each position in A equals element at corresponding position in B. • Matrices having equal corresponding entries. • Examples 5 0  4 4 3 4          5 0 1 0.75       Lecture Slides By Adil Aslam 9
  10. 10. Matrix Operations • Matrix Addition • Let A = [aij] and B = [bij] be m x n matrices. The sum of A and B, denoted by A + B, is the m x n matrix that has aij + bij as its (i, j)th element. In other words, A + B = [aij + bij]. If and then A  a11 a12 a21 a22     B  b11 b12 b21 b22     C  a11  b11 a12  b12 a21  b21 a22  b22     Lecture Slides By Adil Aslam 10
  11. 11. Matrix Operations • Matrix Addition • Example: 5 22 3-1-3 24 -4 211 03-1 14 -3 043 3-22 10 -1                                  Lecture Slides By Adil Aslam 11
  12. 12. Matrix Operations • Properties Lecture Slides By Adil Aslam 12 Matrices A, B and C are conformable, A + B = B + A A + (B +C) = (A + B) +C l(A + B) = lA + lB, where l is a scalar (commutative law) (associative law) (distributive law)
  13. 13. Matrix Operations • Subtracting Matrices • Matrices of same dimension can be subtracted. • For example Lecture Slides By Adil Aslam 13
  14. 14. Your Task! Find the Following….. Lecture Slides By Adil Aslam 14 1 2 3 0 1 4        A 2 3 0 1 2 5       BExample: if and Evaluate A + B and A – B. 1 2 2 3 3 0 3 5 3 0 ( 1) 1 2 4 5 1 3 9                    A B 1 2 2 3 3 0 1 1 3 0 ( 1) 1 2 4 5 1 1 1                       A B
  15. 15. Matrix Operations • Matrix Multiplication • Let A be an m x k matrix and B be a k x n matrix. The product of A and B, denoted by AB, is the m x n matrix with its (i, j)th entry equal to the sum of the products of the corresponding elements from the ith row of A and the jth column of B. In other words, if AB = [cij], then • Cij = ai1b1j + ai2b2j + … + aikbkj. • Multiply rows times columns • You can only multiply if the number of columns in the 1st matrix is equal to the number of rows in the 2nd matrix. Lecture Slides By Adil Aslam 15
  16. 16. Matrix Operations Example: Dimensions: 2 x 3 3 x 2 The dimensions of your answer. They must match Lecture Slides By Adil Aslam 16
  17. 17. Matrix Operations • Examples: 1. 2 1 3 4        3 9 2 5 7 6       2(3) + -1(5) 2(-9) + -1(7) 2(2) + -1(-6) 3(3) +4(5) 3(-9) +4(7) 3(2) + 4(-6) 1 25 10 29 1 18       Lecture Slides By Adil Aslam 17
  18. 18. Matrix Operations 0 1 4 3 1 0 2 5             4. 2 x 2 2 x 2 *Answer should be a 2 x 2 0(4) + (-1)(-2) 0(-3) + (-1)(5) 1(4) + 0(-2) 1(-3) +0(5)  2 -5 4 -3       Lecture Slides By Adil Aslam 18
  19. 19. Matrix Operations • Example: Let • Does AB = BA? • Solution: We find that • Hence, AB  BA. • Because Matrix multiplication is not commutative. 11 12 Band 12 11 A              23 4 3 BAand 5 3 23 AB              Lecture Slides By Adil Aslam 19
  20. 20. Simplify • Example Find a.) A(B+C) b.) AB+AC                     23 11 , 24 02 , 31 12 CBA Lecture Slides By Adil Aslam 20
  21. 21. Solution A(B+C) Lecture Slides By Adil Aslam 21
  22. 22. Solution AB+AC  2 1 1 3       2 0 4 2       2 1 1 3       1 1 3 2        0 2 14 6       5 4 8 5        5 6 22 11       Lecture Slides By Adil Aslam 22
  23. 23. Your Task! • Matrix multiplication Lecture Slides By Adil Aslam 23 1 2 3 0 1 4        A 1 2 2 3 5 0          BExample: , , Evaluate C = AB. 11 12 21 22 1 ( 1) 2 2 3 5 18 1 2 1 2 2 3 3 0 81 2 3 2 3 0 ( 1) 1 2 4 5 220 1 4 5 0 0 2 1 3 4 0 3 c c c c                                               1 2 1 2 3 18 8 2 3 0 1 4 22 3 5 0 C AB                   
  24. 24. Matrix Operations • Properties Lecture Slides By Adil Aslam 24 Matrices A, B and C are conformable, A(B + C) = AB + AC (A + B)C = AC + BC A(BC) = (AB) C
  25. 25. Transposes and Power Matrices • Identity Matrix • The identity matrix of order n is the n x n matrix In = [ij], where ij = 1 if i = j and ij = 0 if i  j. Hence Lecture Slides By Adil Aslam 25
  26. 26. Transposes and Power Matrices • Identity Matrix • An identity matrix is a diagonal matrix where the diagonal elements all equal one. I= A  I = A             1000 0100 0010 0001 Lecture Slides By Adil Aslam 26
  27. 27. Transposes and Power Matrices • Identity Matrix Example 22 identity matrix 33 identity matrix Lecture Slides By Adil Aslam 27
  28. 28. Transposes and Power Matrices • Transposes of a Matrix • Let A = [aij] be an m x n matrix. The transpose of A, denoted At, is the n x m matrix obtained by interchanging the rows and the columns of A. In other words, if At = [bij], then bij = aij for i = 1, 2, …, n and j = 1, 2, …, m. • The transpose of a matrix is a new matrix that is formed by interchanging the rows and columns. • The transpose of A is denoted by A' or (AT) Lecture Slides By Adil Aslam 28
  29. 29. Transposes and Power Matrices • Transposes of a Matrix • Thus, • If A = A', then A is symmetric          3231 2221 1211 aa aa aa A      322212 312111 aaa aaa A' Lecture Slides By Adil Aslam 29
  30. 30. Transposes and Power Matrices • Example • The transpose of the matrix is Lecture Slides By Adil Aslam 30
  31. 31. Null Matrix • A square matrix where all elements equal zero. Lecture Slides By Adil Aslam 31
  32. 32. Symmetric Matrix • Symmetric Matrix • A square matrix A is called symmetric if A = At. Thus A = [aij] is symmetric if aij = aji for all i and j with 1  i  n and 1  j  n.  Theorems: • If A and B are n x n symmetric matrices, then (AB)' = BA • If A and B are n x n symmetric matrices, then (A+B)' = B+A • If C is any n x n matrix, then B = C'C is symmetric Example: The matrix is symmetric 010 101 011           Lecture Slides By Adil Aslam 32
  33. 33. Zero-One Matrices •A matrix with entries that are either 0 or 1 is called a zero- one matrix. Zero-one matrices are often used like a “table” to represent discrete structures. •We can define Boolean operations on the entries in zero- one matrices: Lecture Slides By Adil Aslam 33
  34. 34. Zero-One Matrices • We can define Boolean operations on the entries in zero- one matrices: Lecture Slides By Adil Aslam 34
  35. 35. Zero-One Matrices •Let A = [aij] and B = [bij] be mn zero-one matrices. •Then the join of A and B is the zero-one matrix with (i, j)th entry aij  bij. The join of A and B is denoted by A  B. •The meet of A and B is the zero-one matrix with (i, j)th entry aij  bij. The meet of A and B is denoted by A  B. Lecture Slides By Adil Aslam 35
  36. 36. Zero-One Matrices •Example:            01 10 11 A            00 11 10 B Join:                          01 11 11 0001 1110 1101 BA Meet:                          00 10 10 0001 1110 1101 BA Lecture Slides By Adil Aslam 36
  37. 37. Zero-One Matrices •Let A = [aij] be an mk zero-one matrix and B = [bij] be a kn zero-one matrix. •Then the Boolean product of A and B, denoted by AB, is the mn matrix with (i, j)th entry [cij], where cij = (ai1  b1j)  (ai2  b2i)  …  (aik  bkj). •Note that the actual Boolean product symbol has a dot in its center. •Basically, Boolean multiplication works like the multiplication of matrices, but with computing  instead of the product and  instead of the sum. Lecture Slides By Adil Aslam 37
  38. 38. Zero-One Matrices Example:        11 01 A        10 10 B                10 10 )11()11()01()01( )10()11()00()01( BA Lecture Slides By Adil Aslam 38
  39. 39. Zero-One Matrices • Example: Find the Boolean product of A and B, where • Solution: . 110 011 B, 01 10 01 A                                                          011 110 011 000101 101000 000101 )10()01()10()11(0)(01)(1 )11()00()11()10(0)(11)(0 )10()01()10()11(0)(01)(1 BA Lecture Slides By Adil Aslam 39
  40. 40. Zero-One Matrices •Let A be a square zero-one matrix and r be a positive integer. •The r-th Boolean power of A is the Boolean product of r factors of A. The r-th Boolean power of A is denoted by A[r]. •A[0] = In, •A[r] = AA…A (r times the letter A) Lecture Slides By Adil Aslam 40
  41. 41. Properties of Matrix Lecture Slides By Adil Aslam 41 (AB)-1 = B-1A-1 (AT)T = A and (lA)T = lAT (A + B)T = AT + BT (AB)T = BT AT
  42. 42. Lecture Slides By Adil Aslam 42

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Definition Of Matrix Square Matrix with Example Matrix Operations Matrix Multiplication with Example Transposes and Power Matrices Transposes of a Matrix with Example Symmetric Matrix with Example Zero-One Matrices Properties of Matrix

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