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Understanding Matrix Types and Operations: A Comprehensive Guide

This resource provides an in-depth exploration of matrices, including definitions, types, and fundamental operations. It covers essential matrix types such as row, column, square, symmetric, diagonal, and identity matrices. Additionally, it explains matrix operations including addition, subtraction, multiplication, and the use of scalar multiplication. The resource further discusses the application of matrices in solving simultaneous equations, as well as matrix inversion methods and the Gauss-Jordan elimination technique. A must-read for students and professionals in mathematics and engineering.

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Understanding Matrix Types and Operations: A Comprehensive Guide

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  1. Aij i = row j = column Definition of a Matrix A [ A ]

  2. a11 a12 a13 … … a1n a21 a22 a23 … … a2n … … … aij … … am1 am2 am3 … … amn Definition of a Matrix

  3. a11 a12 a13 … … a1n a21 a22 a23 … … a2n … … … aij … … am1 am2 am3 … … amn size m x n Size of a Matrix

  4. Size of a Matrix 5 21 3 -7 40 -6 19 23 -8 12 50 22 size 3 x 4

  5. Row Matrix [ B ] m = 1 [ 50 -3 -27 35 ]

  6. {D} n = 1 -10 33 -6 15 {-10 33 -6 15} Column Matrix

  7. a11 a12 a13 … … a1n a21 a22 a23 … … a2n … … … aij … … an1 an2 an3 … … ann size m x n 5 21 3 40 -6 19 -8 12 50 size 3 x 3 Square Matrix m = n

  8. a11 a12 a13 … … a1n a21 a22 a23 … … a2n … … … aij … … an1 an2 an3 … … ann 5 21 3 40 -6 19 -8 12 50 5, -6, and 50 are diagonal elements Main Diagonal i = j a11 a22 aij, …, …, ann

  9. a11 a12 a13 … … a1n a21 a22 a23 … … a2n … … … aij … … an1 an2 an3 … … ann Symmetric Matrix aij = aji a12 = a21, a13 = a31, … a1n = an1

  10. Symmetric Matrix 5 21 -3 21 6 19 -8 19 50 21, -3, and 19 are off-diagonal elements

  11. Diagonal Matrix aij = 0, for a  j a11 0 0 … … 0 0 a22 0 … … 0 … … … aij … … 0 0 0 … … 0 a12 = a21 = 0, a13 = a31 = 0, … a1n = an1= 0

  12. 0 0 0 • 0 6 0 0 • 0 0 19 0 • 0 0 0 21 Diagonal Matrix

  13. 1 0 0 … … 0 0 1 0 … … 0 … … … aij … … 0 0 0 … … 1 Unit or Identity Matrix aij = 1, for i = j aij = 0, for i  j a12 = a21 = 0, a13 = a31 = 0, … a1n = an1= 0

  14. 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 null matrix aij =0 Unit or Identity Matrix

  15. Matrix Operations

  16. 5 21 -3 A = 21 6 19 -8 19 50 5 21 -3 B = 21 6 19 -8 19 50 Equality A = B Aij = Bij

  17. 5 2 A = 2 6 -8 1 5 21 B = 21 6 -8 19 Addition and Subtraction [A] + [B] = [C] Aij + Bij = Cij

  18. 10 23 A+B = C = 23 12 -16 20 0 -19 A-B = C = -19 0 0 -18 Addition and Subtraction

  19. 5 2 A = 2 6 -8 1 15 6 B = 6 18 -24 3 Multiplication by ScalarScalar c, x [A] c = 3 c A = B

  20. -1 5 2 3 6A = B = 7 -3 4 –8 9 18 -43 51 C = 2 45 -69 Multiplication of Matrices Conformable [A] (m x n) x [B] (n x s) = [C] (m x s) Aik x Bkj = C ij Cij = Ai1B1j +ai2B2j+ … + AinBnj Cij =  AikBkj for k = 1 to n

  21. 2 3 6 B = 4 –8 9 -1 5 18 -43 51 A = C = 7 -3 2 45 -69 Manual Multiplication

  22. Application to Simultaneous Equations a11x1 + a12x2 + a13x3 = P1 a12x2 + a22x2 + a23x3 = P2 a12x3 + a23x2 + a33x3 = P3 2x1 – 5x2 + 4x3 = 44 3x1 + 1x2 + -8x3 = -35 4x1 – 7x2 – 1x3 = 28

  23. Application to Simultaneous Equations a11 a12 a13 x1 P1 a12 a22 a23 x2 = P2 a12 a23 a33 x3 P3

  24. 2 -5 4 x1 44 3 1 -8 x2 = -35 4 -7 -1 x3 -28 [A] {x} = {P} NOTES: [A] [B]  [B] [A] A B C = (AB) C = A (BC) A (B + C) = AB + AC [A] [0] = [0], [0] [A] = [0] Application to Simultaneous Equations

  25. 1 -2 A = 3 4 -2 1 A-1 = -1.5 0.5 Inverse of a Square Matrix Inverse of [A] = [A-1] [A-1] [A] = [I][A] [A-1] = [I]

  26. Inverse of Square Matrix 1 0 A A-1 = 0 1

  27. Transpose of a Matrix aijT = aji a11 a21 a31 … … an1 a12 a22 a32 … … an2 … … … aji … … a1n a2n a3n … … ann

  28. 5 12 -3 18 21 6 19 16 -3 15 50 17 5 21 -3 12 6 15 -3 19 50 18 16 17 Transpose of a Matrix A (3 x 4) , AT(4 x 3)

  29. 3 5 -1 ¦ 2 -2 4 7 ¦ 9 6 1 3 ¦ 4 1 8 -5 2 -3 6 7 -1 Partitioning of Matrices [A] [B]

  30. Partitioning of Matrices A11 ¦ A12 A = -----¦------- A21 ¦ A22 B = B11 ------ B21

  31. Partitioning of Matrices B11 B = ------ B21 A11 | A12 A11B11+A12B21 A= ---------------- AB= A21 | A22 A21B11+A22B21

  32. 19 28 A11B11 = -43 34 14 -2 A12B21 = 63 -9 Partitioning of Matrices A21B11 = [ -8 68 ] A22B21 = [ 28 -4 ]

  33. Partitioning of Matrices A11B11+A12B21 AB = A21B11+A22B21 19 28 + 14 -2 -6 26 AB = -43 34 + 63 -9 = 20 25 [-8 68 ] + [28 -4] 20 64

  34. Solution of Simultaneous Equations by Gauss-Jordan Method 2x1 – 5x2 + 4x3 = 44 3x1 + x2 - 9x3 = -35 4x1 – 7x2 - x3 = 28 x1 – 2.5x2 + 2x3 = 22 3x1 + x2 - 8x3 = -35 4x1 - 7x2 - x3 = 28

  35. Solution of Simultaneous Equations by Gauss-Jordan Method x1 – 2.5x2 + 2x3 = 22 8.5x2 - 14x3 = -101 3x2 - 9x3 = -60 x1 – 2.5x2 + 2x3 = 22 x2 - 1.647x3 = -11.882 3x2 - 9x3 = -60

  36. Solution of Simultaneous Equations by Gauss-Jordan Method x1 – - 2.118x3 = -7.705 x2 - 1.647x3 = -11.882 - 4.059x3 = -24.354 x1 + 2.118x3 = - 7.705 x2 - 1.647x3 = -11.882 x3 = 6 x1 = 5 x2 = -2 x3 = 6

  37. Solution of Simultaneous Equations by Gauss-Jordan Method Check: 2(5) - 5(-2) + 4(6) = 44 3(5) +1(-2) - 8(6) = -35 4(5) - 7(-2) - 1(6) = 28

  38. Matrix Inversion [A] {x} = {C} [A] [A] {x} = [A]-1 {C} [A] [A] = [I] {x} = [A] {C} [A ¦ I ] { x ¦ -C }= 0 -1 -1 -1

  39. Matrix Inversion [I ¦ B ] { x ¦ -C }= 0 {x} - [B] [C] = 0 {x} = [B] [C] [B] = [A] -1

  40. Method of Successive Transformations 2 4 3 ¦ 1 0 0 1 -2 0 ¦ 0 1 0 -1 -4 5 ¦ 0 0 1 1 2 1.5 ¦ 0.5 0 0 1 -2 0 ¦ 0 1 0 -1 -4 5 ¦ 0 0 1

  41. Method of Successive Transformations 1 2 1.5 ¦ 0.5 0 0 0 -4 -1.5 ¦ -0.5 1 0 -1 -4 5 ¦ 0 0 1 1 2 1.5 ¦ 0.5 0 0 0 -4 -1.5 ¦ -0.5 1 0 0 -2 6.5 ¦ 0.5 0 1

  42. Method of Successive Transformations 1 2 1.5 ¦ 0.5 0 0 0 1 0.375 ¦ 0.125 -0.25 0 0 -2 6.5 ¦ 0.5 0 1 1 2 1.5 ¦ 0.5 0 0 0 1 0.375 ¦ 0.125 -0.25 0 0 0 7.25 ¦ 0.75 -0.5 1

  43. Method of Successive Transformations 1 2 1.5 ¦ 0.5 0 0 0 1 0.375 ¦ 0.125 -0.25 0 0 0 1 ¦ 0.1034 -0.06897 0.1379 1 2 1.5 ¦ 0.5 0 0 0 1 0 ¦ 0.0862 -0.2241 -0.0517 0 0 1 ¦ 0.1034 -0.06897 0.1379

  44. Method of Successive Transformations 1 2 0 ¦ 0.3449 0.1034 -0.2069 0 1 0 ¦ 0.0862 -0.2241 -0.0517 0 0 1 ¦ 0.1034 -0.06897 0.1379 1 0 0 ¦ 0.1725 0.5516 -0.1035 0 1 0 ¦ 0.0862 -0.2241 -0.0517 0 0 1 ¦ 0.1034 -0.06897 0.1379

  45. Method of Successive Transformations 0.1725 0.5516 - 0.1035 A-1 = 0.0862 - 0.2241 - 0.0517 0.1034 - 0.06897 0.1379

  46. Cholesky Decomposition Lower Triangular matrix [L] l11 0 0 . . . . 0 l21 l22 0 . . . . 0 l31 l32 l33 0 . . . 0 . . . . . . . . . . . . . . . . ln1 . . . . . . lnn

  47. Cholesky Decomposition [A] = [L] [L]T [B] = [L] [A] = ( [L] [L] ) [A] = [B] [B] -1 -1 T -1 -1 T

  48. Cholesky Decomposition Elements of [L]: l = 0 for i<j l = (A - ∑l ) l = (A - ∑l l )/l for i>j Summation ∑ from r=1 to j-1 ij 2 1/2 ir ij ii ij ij ir jr

  49. Cholesky Decomposition Elements of [B]: b = 0 for i<j b = 1/l b = -(∑l l )/l or i>j Summation ∑ from r=1 to i-1 ij ii ii ij rj ii ir

  50. Cholesky Decomposition Example: 2 1 1 1 1.5 2 1 2 6.75

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