1. Matrix Multiplication
Example 1
Original author: Jeffrey Bivin, Lake Zurich High School

2. Matrix Multiplication3 2 5 -2 1 4 6 9 5 -1 2 3 7 4 X
Row by Column
Original author: Jeffrey Bivin, Lake Zurich High School

3. Row 1 x Column 1 3 2 5 -2 1 4 6 9 5 -1 2 3 7 4 X
3 x x x 2 = 25
Original author: Jeffrey Bivin, Lake Zurich High School

4. Row 1 x Column 1 3 2 5 -2 1 4 6 9 5 -1 2 3 7 4 X
3 x x x 2 = 25
Original author: Jeffrey Bivin, Lake Zurich High School

5. Row 1 x Column 1 3 2 5 -2 1 4 6 9 5 -1 2 3 7 4 X
3 x x x 2 = 25
Original author: Jeffrey Bivin, Lake Zurich High School

6. Row 1 x Column 1 3 2 5 -2 1 4 6 9 5 -1 2 3 7 4 X 25 =
A(1,1)
Original author: Jeffrey Bivin, Lake Zurich High School

7. 3 x (-1) + 2 x 3 + 5 x 7 = 38 X Row 1 x Column 2 3 2 5 -2 1 4 6 9 5 -11 4 6 9 5 -1 2 3 7 4 X
3 x (-1) + 2 x x 7 = 38
Original author: Jeffrey Bivin, Lake Zurich High School

8. 3 x (-1) + 2 x 3 + 5 x 7 = 38 X Row 1 x Column 2 3 2 5 -2 1 4 6 9 5 -11 4 6 9 5 -1 2 3 7 4 X
3 x (-1) + 2 x x 7 = 38
Original author: Jeffrey Bivin, Lake Zurich High School

9. 3 x (-1) + 2 x 3 + 5 x 7 = 38 X Row 1 x Column 2 3 2 5 -2 1 4 6 9 5 -11 4 6 9 5 -1 2 3 7 4 X
3 x (-1) + 2 x x 7 = 38
Original author: Jeffrey Bivin, Lake Zurich High School

10. Row 1 x Column 2 3 2 5 -2 1 4 6 9 5 -1 2 3 7 4 X 25 38 =
A(1,2)
Original author: Jeffrey Bivin, Lake Zurich High School

11. Row 1 x Column 3 3 2 5 -2 1 4 6 9 5 -1 2 3 7 4 X
3 x x x 4 = 40
Original author: Jeffrey Bivin, Lake Zurich High School

12. Row 1 x Column 3 3 2 5 -2 1 4 6 9 5 -1 2 3 7 4 X
3 x x x 4 = 40
Original author: Jeffrey Bivin, Lake Zurich High School

13. Row 1 x Column 3 3 2 5 -2 1 4 6 9 5 -1 2 3 7 4 X
3 x x x 4 = 40
Original author: Jeffrey Bivin, Lake Zurich High School

14. Row 1 x Column 3 3 2 5 -2 1 4 6 9 5 -1 2 3 7 4 X 25 38 40 =
A(1,3)
Original author: Jeffrey Bivin, Lake Zurich High School

15. -2 x 5 + 0 x 0 + 1 x 2 = -8 X Row 2 x Column 1 3 2 5 -2 1 4 6 9 5 -1 21 4 6 9 5 -1 2 3 7 4 X
-2 x x x 2 = -8
Original author: Jeffrey Bivin, Lake Zurich High School

16. -2 x 5 + 0 x 0 + 1 x 2 = -8 X Row 2 x Column 1 3 2 5 -2 1 4 6 9 5 -1 21 4 6 9 5 -1 2 3 7 4 X
-2 x x x 2 = -8
Original author: Jeffrey Bivin, Lake Zurich High School

17. -2 x 5 + 0 x 0 + 1 x 2 = -8 X Row 2 x Column 1 3 2 5 -2 1 4 6 9 5 -1 21 4 6 9 5 -1 2 3 7 4 X
-2 x x x 2 = -8
Original author: Jeffrey Bivin, Lake Zurich High School

18. Row 2 x Column 1 3 2 5 -2 1 4 6 9 5 -1 2 3 7 4 X 25 38 40 -8 =
A(2,1)
Original author: Jeffrey Bivin, Lake Zurich High School

19. Row 2 x Column 2 3 2 5 -2 1 4 6 9 5 -1 2 3 7 4 X
Original author: Jeffrey Bivin, Lake Zurich High School

20. -2 x (-1) + 0 x 3 + 1 x 7 = 9 X Row 2 x Column 2 3 2 5 -2 1 4 6 9 5 -11 4 6 9 5 -1 2 3 7 4 X
-2 x (-1) + 0 x x 7 = 9
Original author: Jeffrey Bivin, Lake Zurich High School

21. -2 x (-1) + 0 x 3 + 1 x 7 = 9 X Row 2 x Column 2 3 2 5 -2 1 4 6 9 5 -11 4 6 9 5 -1 2 3 7 4 X
-2 x (-1) + 0 x x 7 = 9
Original author: Jeffrey Bivin, Lake Zurich High School

22. -2 x (-1) + 0 x 3 + 1 x 7 = 9 X Row 2 x Column 2 3 2 5 -2 1 4 6 9 5 -11 4 6 9 5 -1 2 3 7 4 X
-2 x (-1) + 0 x x 7 = 9
Original author: Jeffrey Bivin, Lake Zurich High School

23. Row 2 x Column 2 3 2 5 -2 1 4 6 9 5 -1 2 3 7 4 X 25 38 40 -8 9 =
A(2,2)
Original author: Jeffrey Bivin, Lake Zurich High School

24. -2 x 2 + 0 x 7 + 1 x 4 = 0 X Row 2 x Column 3 3 2 5 -2 1 4 6 9 5 -1 21 4 6 9 5 -1 2 3 7 4 X
-2 x x x 4 = 0
Original author: Jeffrey Bivin, Lake Zurich High School

25. -2 x 2 + 0 x 7 + 1 x 4 = 0 X Row 2 x Column 3 3 2 5 -2 1 4 6 9 5 -1 21 4 6 9 5 -1 2 3 7 4 X
-2 x x x 4 = 0
Original author: Jeffrey Bivin, Lake Zurich High School

26. -2 x 2 + 0 x 7 + 1 x 4 = 0 X Row 2 x Column 3 3 2 5 -2 1 4 6 9 5 -1 21 4 6 9 5 -1 2 3 7 4 X
-2 x x x 4 = 0
Original author: Jeffrey Bivin, Lake Zurich High School

27. Row 2 x Column 3 3 2 5 -2 1 4 6 9 5 -1 2 3 7 4 X 25 38 40 -8 9 =
A(2,3)
Original author: Jeffrey Bivin, Lake Zurich High School

28. Row 3 x Column 1 3 2 5 -2 1 4 6 9 5 -1 2 3 7 4 X
4 x x x 2 = 38
Original author: Jeffrey Bivin, Lake Zurich High School

29. Row 3 x Column 1 3 2 5 -2 1 4 6 9 5 -1 2 3 7 4 X
4 x x x 2 = 38
Original author: Jeffrey Bivin, Lake Zurich High School

30. Row 3 x Column 1 3 2 5 -2 1 4 6 9 5 -1 2 3 7 4 X
4 x x x 2 = 38
Original author: Jeffrey Bivin, Lake Zurich High School

31. Row 3 x Column 1 3 2 5 -2 1 4 6 9 5 -1 2 3 7 4 X 25 38 40 -8 9 =
A(3,1)
Original author: Jeffrey Bivin, Lake Zurich High School

32. 4 x (-1) + 6 x 3 + 9 x 7 = 77 X Row 3 x Column 2 3 2 5 -2 1 4 6 9 5 -11 4 6 9 5 -1 2 3 7 4 X
4 x (-1) + 6 x x 7 = 77
Original author: Jeffrey Bivin, Lake Zurich High School

33. 4 x (-1) + 6 x 3 + 9 x 7 = 77 X Row 3 x Column 2 3 2 5 -2 1 4 6 9 5 -11 4 6 9 5 -1 2 3 7 4 X
4 x (-1) + 6 x x 7 = 77
Original author: Jeffrey Bivin, Lake Zurich High School

34. 4 x (-1) + 6 x 3 + 9 x 7 = 77 X Row 3 x Column 2 3 2 5 -2 1 4 6 9 5 -11 4 6 9 5 -1 2 3 7 4 X
4 x (-1) + 6 x x 7 = 77
Original author: Jeffrey Bivin, Lake Zurich High School

35. Row 3 x Column 2 3 2 5 -2 1 4 6 9 5 -1 2 3 7 4 X 25 38 40 -8 9 77 =
A(3,2)
Original author: Jeffrey Bivin, Lake Zurich High School

36. Row 3 x Column 3 3 2 5 -2 1 4 6 9 5 -1 2 3 7 4 X
4 x x x 4 = 86
Original author: Jeffrey Bivin, Lake Zurich High School

37. Row 3 x Column 3 3 2 5 -2 1 4 6 9 5 -1 2 3 7 4 X
4 x x x 4 = 86
Original author: Jeffrey Bivin, Lake Zurich High School

38. Row 3 x Column 3 3 2 5 -2 1 4 6 9 5 -1 2 3 7 4 X
4 x x x 4 = 86
Original author: Jeffrey Bivin, Lake Zurich High School

39. Row 3 x Column 3 3 2 5 -2 1 4 6 9 5 -1 2 3 7 4 X 25 38 40 -8 9 77 86
A(3,3)
Original author: Jeffrey Bivin, Lake Zurich High School

40. Matrix Multiplication3 2 5 -2 1 4 6 9 5 -1 2 3 7 4 X 25 38 40 -8 9 77 86 =
Original author: Jeffrey Bivin, Lake Zurich High School

41. Matrix Multiplication
Example 2
Original author: Jeffrey Bivin, Lake Zurich High School

42. Matrix Multiplication3 5 -2 1 4 9 5 2 3 4 X
Row by Column
Original author: Jeffrey Bivin, Lake Zurich High School

43. Row 1 X Column 1 3 5 -2 1 4 9 5 2 3 4 X
3 x x 3 = 30
Original author: Jeffrey Bivin, Lake Zurich High School

44. Row 1 X Column 1 3 5 -2 1 4 9 5 2 3 4 X
3 x x 3 = 30
Original author: Jeffrey Bivin, Lake Zurich High School

45. Row 1 X Column 1 3 5 -2 1 4 9 5 2 3 4 X 30 =
A(1,1)
Original author: Jeffrey Bivin, Lake Zurich High School

46. Row 1 X Column 2 3 5 -2 1 4 9 5 2 3 4 X
3 x x 4 = 26
Original author: Jeffrey Bivin, Lake Zurich High School

47. Row 1 X Column 2 3 5 -2 1 4 9 5 2 3 4 X
3 x x 4 = 26
Original author: Jeffrey Bivin, Lake Zurich High School

48. Row 1 X Column 2 3 5 -2 1 4 9 5 2 3 4 X 30 26 =
A(1,2)
Original author: Jeffrey Bivin, Lake Zurich High School

49. Row 2 X Column 1 3 5 -2 1 4 9 5 2 3 4 X
-2 x x 3 = -7
Original author: Jeffrey Bivin, Lake Zurich High School

50. Row 2 X Column 1 3 5 -2 1 4 9 5 2 3 4 X
-2 x x 3 = -7
Original author: Jeffrey Bivin, Lake Zurich High School

51. Row 2 X Column 1 3 5 -2 1 4 9 5 2 3 4 X 30 26 -7 =
A(2,1)
Original author: Jeffrey Bivin, Lake Zurich High School

52. Row 2 X Column 2 3 5 -2 1 4 9 5 2 3 4 X
-2 x x 4 = 0
Original author: Jeffrey Bivin, Lake Zurich High School

53. Row 2 X Column 2 3 5 -2 1 4 9 5 2 3 4 X
-2 x x 4 = 0
Original author: Jeffrey Bivin, Lake Zurich High School

54. Row 2 X Column 2 3 5 -2 1 4 9 5 2 3 4 X 30 26 -7 =
A(2,2)
Original author: Jeffrey Bivin, Lake Zurich High School

55. Row 3 X Column 1 3 5 -2 1 4 9 5 2 3 4 X
4 x x 3 = 47
Original author: Jeffrey Bivin, Lake Zurich High School

56. Row 3 X Column 1 3 5 -2 1 4 9 5 2 3 4 X
4 x x 3 = 47
Original author: Jeffrey Bivin, Lake Zurich High School

57. Row 3 X Column 1 3 5 -2 1 4 9 5 2 3 4 X 30 26 -7 47 =
A(3,1)
Original author: Jeffrey Bivin, Lake Zurich High School

58. Row 3 X Column 2 3 5 -2 1 4 9 5 2 3 4 X
4 x x 4 = 44
Original author: Jeffrey Bivin, Lake Zurich High School

59. Row 3 X Column 2 3 5 -2 1 4 9 5 2 3 4 X
4 x x 4 = 44
Original author: Jeffrey Bivin, Lake Zurich High School

60. Row 3 X Column 2 3 5 -2 1 4 9 5 2 3 4 X 30 26 -7 47 44 =
A(3,2)
Original author: Jeffrey Bivin, Lake Zurich High School

61. Matrix Multiplication3 5 -2 1 4 9 5 2 3 4 X 30 26 -7 47 44 =
Original author: Jeffrey Bivin, Lake Zurich High School

62. Matrix Multiplication
Example 3
Original author: Jeffrey Bivin, Lake Zurich High School

63. Row 1 x Column 1 5 3 -2 4 -1 2 1 3 4 5 6 X
5 x x 4 = 22
Original author: Jeffrey Bivin, Lake Zurich High School

64. Row 1 x Column 1 5 3 -2 4 -1 2 1 3 4 5 6 X
5 x x 4 = 22
Original author: Jeffrey Bivin, Lake Zurich High School

65. Row 1 x Column 1 5 3 -2 4 -1 2 1 3 4 5 6 X 22 =
A(1,1)
Original author: Jeffrey Bivin, Lake Zurich High School

66. Row 1 x Column 2 5 3 -2 4 -1 2 1 3 4 5 6 X
5 x x 4 = 22
Original author: Jeffrey Bivin, Lake Zurich High School

67. Row 1 x Column 2 5 3 -2 4 -1 2 1 3 4 5 6 X
5 x x 5 = 15
Original author: Jeffrey Bivin, Lake Zurich High School

68. Row 1 x Column 2 5 3 -2 4 -1 2 1 3 4 5 6 X 22 15 =
A(1,2)
Original author: Jeffrey Bivin, Lake Zurich High School

69. Row 1 x Column 3 5 3 -2 4 -1 2 1 3 4 5 6 X
5 x x 5 = 15
Original author: Jeffrey Bivin, Lake Zurich High School

70. Row 1 x Column 3 5 3 -2 4 -1 2 1 3 4 5 6 X
5 x x 2 = 11
Original author: Jeffrey Bivin, Lake Zurich High School

71. Row 1 x Column 3 5 3 -2 4 -1 2 1 3 4 5 6 X 22 15 11 =
A(1,3)
Original author: Jeffrey Bivin, Lake Zurich High School

72. Row 1 x Column 4 5 3 -2 4 -1 2 1 3 4 5 6 X
5 x x 6 = 33
Original author: Jeffrey Bivin, Lake Zurich High School

73. Row 1 x Column 4 5 3 -2 4 -1 2 1 3 4 5 6 X
5 x x 6 = 33
Original author: Jeffrey Bivin, Lake Zurich High School

74. Row 1 x Column 4 5 3 -2 4 -1 2 1 3 4 5 6 X 22 15 11 33 =
A(1,4)
Original author: Jeffrey Bivin, Lake Zurich High School

75. Row 2 x Column 1 5 3 -2 4 -1 2 1 3 4 5 6 X
-2 x x 4 = 12
Original author: Jeffrey Bivin, Lake Zurich High School

76. Row 2 x Column 1 5 3 -2 4 -1 2 1 3 4 5 6 X
-2 x x 4 = 12
Original author: Jeffrey Bivin, Lake Zurich High School

77. Row 2 x Column 1 5 3 -2 4 -1 2 1 3 4 5 6 X 22 15 11 33 12 =
A(2,1)
Original author: Jeffrey Bivin, Lake Zurich High School

78. Row 2 x Column 2 5 3 -2 4 -1 2 1 3 4 5 6 X
-2 x x 5 = 20
Original author: Jeffrey Bivin, Lake Zurich High School

79. Row 2 x Column 2 5 3 -2 4 -1 2 1 3 4 5 6 X
-2 x x 5 = 20
Original author: Jeffrey Bivin, Lake Zurich High School

80. Row 2 x Column 2 5 3 -2 4 -1 2 1 3 4 5 6 X 22 15 11 33 12 20 =
A(2,2)
Original author: Jeffrey Bivin, Lake Zurich High School

81. Row 2 x Column 3 5 3 -2 4 -1 2 1 3 4 5 6 X
-2 x x 2 = 6
Original author: Jeffrey Bivin, Lake Zurich High School

82. Row 2 x Column 3 5 3 -2 4 -1 2 1 3 4 5 6 X
-2 x x 2 = 6
Original author: Jeffrey Bivin, Lake Zurich High School

83. Row 2 x Column 3 5 3 -2 4 -1 2 1 3 4 5 6 X 22 15 11 33 12 20 6 =
A(2,3)
Original author: Jeffrey Bivin, Lake Zurich High School

84. Row 2 x Column 4 5 3 -2 4 -1 2 1 3 4 5 6 X
-2 x x 6 = 18
Original author: Jeffrey Bivin, Lake Zurich High School

85. Row 2 x Column 4 5 3 -2 4 -1 2 1 3 4 5 6 X
-2 x x 6 = 18
Original author: Jeffrey Bivin, Lake Zurich High School

86. Row 2 x Column 4 5 3 -2 4 -1 2 1 3 4 5 6 X 22 15 11 33 12 20 6 18 =
A(2,4)
Original author: Jeffrey Bivin, Lake Zurich High School

87. Row 3 x Column 1 5 3 -2 4 -1 2 1 3 4 5 6 X
-1 x x 4 = -2
Original author: Jeffrey Bivin, Lake Zurich High School

88. Row 3 x Column 1 5 3 -2 4 -1 2 1 3 4 5 6 X
-1 x x 4 = -2
Original author: Jeffrey Bivin, Lake Zurich High School

89. Row 3 x Column 1 5 3 -2 4 -1 2 1 3 4 5 6 X 22 15 11 33 12 20 6 18 -2 =
A(3,1)
Original author: Jeffrey Bivin, Lake Zurich High School

90. Row 3 x Column 2 5 3 -2 4 -1 2 1 3 4 5 6 X
-1 x x 5 = 0
Original author: Jeffrey Bivin, Lake Zurich High School

91. Row 3 x Column 2 5 3 -2 4 -1 2 1 3 4 5 6 X
-1 x x 5 = 0
Original author: Jeffrey Bivin, Lake Zurich High School

92. Row 3 x Column 2 5 3 -2 4 -1 2 1 3 4 5 6 X 22 15 11 33 12 20 6 18 -2 =
A(3,2)
Original author: Jeffrey Bivin, Lake Zurich High School

93. Row 3 x Column 3 5 3 -2 4 -1 2 1 3 4 5 6 X
-1 x x 2 = -1
Original author: Jeffrey Bivin, Lake Zurich High School

94. Row 3 x Column 3 5 3 -2 4 -1 2 1 3 4 5 6 X
-1 x x 2 = -1
Original author: Jeffrey Bivin, Lake Zurich High School

95. Row 3 x Column 3 5 3 -2 4 -1 2 1 3 4 5 6 X 22 15 11 33 12 20 6 18 -2 -1 =
A(3,3)
Original author: Jeffrey Bivin, Lake Zurich High School

96. Row 3 x Column 4 5 3 -2 4 -1 2 1 3 4 5 6 X
-1 x x 6 = -3
Original author: Jeffrey Bivin, Lake Zurich High School

97. Row 3 x Column 4 5 3 -2 4 -1 2 1 3 4 5 6 X
-1 x x 6 = -3
Original author: Jeffrey Bivin, Lake Zurich High School

98. Row 3 x Column 4 5 3 -2 4 -1 2 1 3 4 5 6 X 22 15 11 33 12 20 6 18 -2 -1 -3 =
A(3,4)
Original author: Jeffrey Bivin, Lake Zurich High School

99. Matrix Multiplication5 3 -2 4 -1 2 1 3 4 5 6 X 22 15 11 33 12 20 6 18 -2 -1 -3 =
Original author: Jeffrey Bivin, Lake Zurich High School