1. Chapter 5.7 Properties of Matrices

2. Basic Definitions It is necessary to use capital letters to name matrices. Also, subscript notation is often used to name element of a matrix, as in the following Matrix A.

4. With this notation, the first row, first column element is a 11 (read “a-sub-one-one”); The second row, third column element is a 23 and in general the i th row, j th column element is a ij

5. Certain matrices have special names: because the number of rows is equal to the number of columns. A matrix with just one row is a row matrix, and a matrix with just one column is a column matrix.

6. Two matrices are equal if they are the same size and if the corresponding element, position by position are equal. Using this definition, the matrices are not equal (even though they contain the same elements and are the same size), since the corresponding elements differ.

7. Find the values of the variables for which each statement is true, if possible.

9. Adding Matrices Addition of matrices is defined as follows. It can be shown that matrix addition satisfies the commutative, associative, closure, identity, and inverse properties.

10. Find the sum if possible

13. Special Matrices A matrix containing only zero elements is called a zero matrix. A zero matrix can be written with any size.

14. By the additive inverse property, each real number has an additive inverse: if a is a real number, then there is a real number –a such that a + (-a) = 0 and –a + a = 0

15. Given matrix A, there is a matrix –A such that A + -A = 0 The matrix –A has as elements the additive inverses of the elements of A.

16. For example, if

18. Matrix –A is called the additive inverse, or negative, of matrix A. Every matrix has an additive inverse Subtracting Matrices The real number b is subtracted from the real number a, written a – b, by adding a and the additive inverse of b. That is, a – b = a + (-b)

19. In practice, the difference of two matrices of the same size is found by subtracting corresponding elements.

20. Find the difference if possible

23. Multiplying Matrices In work with matrices, a real number is called a scalar to distinguish it from a matix. The product of a scalar K and a matrix X is the matrix kX, each of whose elements is k times the corresponding element of X.

24. Find the product

27. We have seen how to multiply a real number (scalar) and a matrix. To find the product of two matrices, such as

28. first locate row 1 of A

29. and column 1 of B,

30. Multiply corresponding elements, and find the sum of the products. (-3)(-6) + (4) (2) + (2)(3) 18+ 8 + 6 = 32

31. The result is the first element for row 1, column 1 in the product matrix. (-3)(-6) + (4) (2) + (2)(3) 18+ 8 + 6 = 32

32. Now use row 1 of A and column 2 of B to determine the element in row 1 and column 2 of the product matrix.

33. (-3)(4) + (4) (3) + (2)(-2) -12 + 12 + -4 = -4

34. Now use row 2 of A and column 1 of B to determine the element in row 2 and column 1 of the product matrix.

35. (5)(-6) + (0) (2) + (4)(3) -30 + 0 + 12 = -18

36. Now use row 2 of A and column 2 of B to determine the element in row 2 and column 2 of the product matrix.

37. (5)(4) + (0) (3) + (4)(-2) 20 + 0 + -8 = 12

38. The product matrix can now be written.

40. Can the product AB be calculated? The following diagram shows that AB can be calculated, because the number of columns of A is equal to the number of rows of B. same size

41. If AB can be calculated, how big is it? same size size of AB = 3 x 4

42. If BA can be calculated? BA cannot be calculated? different size

43. Find the product

54. -8 -3 -13 5 13

55. Find the product

56. -8 -3 -13 5 13 2

57. Find the product

58. -8 -3 -13 5 13 2 1

59. Find the product

60. -8 -3 -13 5 13 2 1 12

61. Find the product Since the first matrix is a 2 x 4 and the second matrix is a 2 x 2 the product can not be found.

62. Find the product

80. The products are not commutative.

83. A contractor builds three kinds of houses, models A, B, and C, with a choice of two styles, colonial or ranch. Matrix P shows the number of each kind of house the contractor is planning for a new 100-home subdivision.

84. The amounts for each of the main materials used depend on the style of the house. Colonial Ranch

85. The amounts are shown in the matrix. Concrete Lumber Brick Shingles

86. Concrete is measured here in cubic yards, lumber in 1000 board feet, brick in 1000s, and shingles in 100 square feet. Concrete Lumber Brick Shingles

87. Cost per Unit

88. What is the total cost of materials for all houses of each model? Concrete Lumber Brick Shingles

89. What is the total cost of materials for all houses of each model? Cost

90. How much of each of the four kinds of material must be ordered?

91. What is the total cost of the materials?