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5 Systems and Matrices Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1

5. 7 Properties of Matrices • Basic Definitions • Adding Matrices • Special Matrices • Subtracting Matrices • Multiplying Matrices • Applying Matrix Algebra Copyright © 2013, 2009, 2005 Pearson Education, Inc. 2

Basic Definitions It is customary to use capital letters to name matrices and to use subscript notation to name elements of a matrix, as in the following matrix A. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 3

Basic Definitions Certain matrices have special names. An n x n matrix is a square matrix 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. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 4

Basic Definitions Two matrices are equal if they are the same size and if corresponding elements, position by position, are equal. Using this definition, the matrices and are not equal (even though they contain the same elements and have the same dimension), since the corresponding elements differ. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 5

Example 1 FINDING VALUES TO MAKE TWO MATRICES EQUAL Find the values of the variables for which each statement is true, if possible. (a) Solution From the definition of equality, the only way that the statement can be true is if 2 = x, 1 = y, p = – 1 and q = 0. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 6

Example 1 FINDING VALUES TO MAKE TWO MATRICES EQUAL Find the values of the variables for which each statement is true, if possible. (b) Solution This statement can never be true since the two matrices have different dimensions. (One is 2 1 and the other is 3 1. ) Copyright © 2013, 2009, 2005 Pearson Education, Inc. 7

Example 2 ADDING MATRICES Find each sum, if possible. (a) Solution Copyright © 2013, 2009, 2005 Pearson Education, Inc. 9

Example 2 ADDING MATRICES Find each sum, if possible. (b) Solution Copyright © 2013, 2009, 2005 Pearson Education, Inc. 10

ADDING MATRICES Example 2 Find each sum, if possible. (c) A + B, if A = and B = Solution The matrices and have different dimension so A and B cannot be added. The sum A + B does not exist. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 11

Special Matrices A matrix containing only zero elements is called a zero matrix. A zero matrix can be written with any dimension. 1 3 zero matrix 2 3 zero matrix Copyright © 2013, 2009, 2005 Pearson Education, Inc. 12

Special Matrices 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 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 13

Special Matrices Given matrix A, there is a matrix – A such that A + (– A) = . The matrix – A has as elements the additive inverses of the elements of A. (Remember, each element of A is a real number and therefore has an additive inverse. ) For example, if Copyright © 2013, 2009, 2005 Pearson Education, Inc. 14

Special Matrices CHECK Test that A + (– A) equals the zero matrix, O. Matrix – A is called the additive inverse, or negative, of matrix A. Every matrix has an additive inverse. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 15

Subtraction of Matrices If A and B are two matrices of the same dimension, then A – B = A + (– B). Copyright © 2013, 2009, 2005 Pearson Education, Inc. 16

Example 3 SUBTRACTING MATRICES Find each difference, if possible. (a) Solution Copyright © 2013, 2009, 2005 Pearson Education, Inc. 17

Example 3 SUBTRACTING MATRICES Find each difference, if possible. (b) Solution Copyright © 2013, 2009, 2005 Pearson Education, Inc. 18

SUBTRACTING MATRICES Example 3 Find each difference, if possible. (c) A – B, if A = and B = Solution The matrices and have different dimensions and cannot be subtracted, so the difference A – B does not exist. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 19

Example 4 MULTIPLYING MATRICES BY SCALARS Find each product. (a) Solution Multiply each element of the matrix by the scalar 5. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 20

Example 4 MULTIPLYING MATRICES BY SCALARS Find each product. (b) Solution Multiply each element of the matrix by the scalar Copyright © 2013, 2009, 2005 Pearson Education, Inc. 21

Properties of Scalar Multiplication If A and B are matrices of the same dimension, and let c and d be scalars. Then these properties hold. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 22

Matrix Multiplication We have seen how to multiply a real number (scalar) and a matrix. To find the product of two matrices, such as first locate row 1 of A and column 1 of B, which are shown shaded below. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 23

Matrix Multiplication Multiply corresponding elements, and find the sum of the products. This result is the element for row 1, column 1 of the product matrix. Now use row 1 of A and column 2 of B to determine the element in row 1, column 2 of the product matrix. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 24

Matrix Multiplication Next, use row 2 of A and column 1 of B. This will give the row 2, column 1 entry of the product matrix. Finally, use row 2 of A and column 2 of B to find the entry for row 2, column 2 of the product matrix. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 25

Matrix Multiplication The product matrix can now be written. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 26

Matrix Multiplication To find the ith row, jth column element of AB, multiply each element in the ith row of A by the corresponding element in the jth column of B. (Note the shaded areas in the matrices below. ) The sum of these products will give the row i, column j element of AB. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 27

Matrix Multiplication The number of columns of an m n matrix A is the same as the number of rows of an n p matrix B (i. e. , both n). The element cij of the product matrix C = AB is found as follows: Matrix AB will be an m p matrix. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 28

Example 5 DECIDING WHETHER TWO MATRICES CAN BE MULTIPLIED Suppose A is a 3 2 matrix, while B is a 2 4 matrix. (a) Can the product AB be calculated? Solution 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. (Both are 2. ) Matrix A Matrix B 3 2 2 4 must match dimension of AB 3 4 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 29

Example 5 DECIDING WHETHER TWO MATRICES CAN BE MULTIPLIED Suppose A is a 3 2 matrix, while B is a 2 4 matrix. (b) If AB can be calculated, what is its dimension? Solution As indicated in the diagram below, the product AB is a 3 x 4 matrix. Matrix A 3 2 Matrix B 2 4 must match dimension of AB 3 4 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 30

Example 5 DECIDING WHETHER TWO MATRICES CAN BE MULTIPLIED Suppose A is a 3 2 matrix, while B is a 2 4 matrix. (c) Can BA be calculated? Solution The diagram below shows that BA cannot be calculated. Matrix B 2 4 Matrix A 3 2 different Copyright © 2013, 2009, 2005 Pearson Education, Inc. 31

Example 5 DECIDING WHETHER TWO MATRICES CAN BE MULTIPLIED Suppose A is a 3 2 matrix, while B is a 2 4 matrix. (d) If BA can be calculated, what is its dimension? Solution The product BA cannot be calculated, because B has 4 columns and A has only 3 rows. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 32

Example 6 MULTIPLYING MATRICES Let Find each product, if possible. (a) AB Solution First decide whether AB can be found. Since A is 2 2 and B is 2 4, the product can be found and will be a 2 4 matrix. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 33

Example 6 MULTIPLYING MATRICES Use the definition of matrix multiplication Perform the operations. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 34

Example 6 MULTIPLYING MATRICES Let Find each product, if possible. (b) BA Solution Since B is a 2 4 matrix, and A is a 2 2 matrix, the number of columns of B (here 4) does not equal the number of rows of A (here 2). Therefore, the product BA cannot be found. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 35

Example 7 MULTIPLYING SQUARE MATRICES IN DIFFERENT ORDERS Let Find each product. (a) AB Solution Copyright © 2013, 2009, 2005 Pearson Education, Inc. 36

Example 7 MULTIPLYING SQUARE MATRICES IN DIFFERENT ORDERS Let Find each product. (b) BA Solution Note that AB ≠ BA. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 37

In general, if A and B are matrices, then AB ≠ BA. Matrix multiplication is not commutative. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 38

Properties of Matrix Multiplication If A, B, and C are matrices such that all the following products and sums exist, then these properties hold. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 39

Example 8 USING MATRIX MULTIPLICATION TO MODEL PLANS FOR A SUBDIVISION 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 to build for a new 100 -home subdivision. The amounts for each of the main materials used depend on the style of the house. These amounts are shown in matrix Q, while matrix R gives the cost in dollars for each kind of material. Concrete is measured here in cubic yards, lumber in 1000 board feet, brick in 1000 s, and shingles in 100 square feet. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 40

Example 8 USING MATRIX MULTIPLICATION TO MODEL PLANS FOR A SUBDIVISION Colonial Ranch Model A Model B Model C Copyright © 2013, 2009, 2005 Pearson Education, Inc. 41

Example 8 USING MATRIX MULTIPLICATION TO MODEL PLANS FOR A SUBDIVISION Concrete Lumber Brick Shingles Colonial Ranch Copyright © 2013, 2009, 2005 Pearson Education, Inc. 42

Example 8 USING MATRIX MULTIPLICATION TO MODEL PLANS FOR A SUBDIVISION Cost per Unit Concrete Lumber Brick Shingles Copyright © 2013, 2009, 2005 Pearson Education, Inc. 43

Example 8 USING MATRIX MULTIPLICATION TO MODEL PLANS FOR A SUBDIVISION (a) What is the total cost of materials for all houses of each model? Solution To find the materials cost for each model, first find matrix PQ, which will show the total amount of each material needed for all houses of each model. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 44

Example 8 Concrete USING MATRIX MULTIPLICATION TO MODEL PLANS FOR A SUBDIVISION Lumber Brick Shingles Model A Model B Model C Copyright © 2013, 2009, 2005 Pearson Education, Inc. 45

Example 8 USING MATRIX MULTIPLICATION TO MODEL PLANS FOR A SUBDIVISION Multiplying PQ and the cost matrix R gives the total cost of materials for each model. Cost Model A Model B Model C Copyright © 2013, 2009, 2005 Pearson Education, Inc. 46

Example 8 USING MATRIX MULTIPLICATION TO MODEL PLANS FOR A SUBDIVISION (b) How much of each of the four kinds of material must be ordered? Solution To find how much of each kind of material to order, refer to the columns of matrix PQ. The sums of the elements of the columns will give a matrix whose elements represent the total amounts of all materials needed for the subdivision. Call this matrix T, and write it as a row matrix. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 47

Example 8 USING MATRIX MULTIPLICATION TO MODEL PLANS FOR A SUBDIVISION (c) What is the total cost of the materials? Solution The total cost of all the materials is given by the product of matrix T, the total amounts matrix, and matrix R, the cost matrix. To multiply these matrices and get a 1 1 matrix, representing the total cost, requires multiplying a 1 4 matrix and a 4 1 matrix. This is why in part (b) a row matrix was written rather than a column matrix. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 48

Example 8 USING MATRIX MULTIPLICATION TO MODEL PLANS FOR A SUBDIVISION The total materials cost is given by TR, so The total cost of materials is \$188, 400. This total may also be found by summing the elements of the column matrix (PQ)R. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 49