Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Chapter 5 More Work with Matrices

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 5.1 Matrix Operations and Solving Matrix Equations Objectives: Use matrix notation Understand what is meant by equal matrices Add/subtract matrices Perform scalar multiplication Solve matrix equations

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall We can represent a matrix in two different ways. 1. A capital letter, such as A, B, or C. 2. A lowercase letter enclosed in brackets, such as that shown below. A [a ij ] A general element in matrix A is denoted by a ij. This refers to the element in the ith row and the jth column. For example, a 32 is the element of A located in the third row, second column. A matrix of order m  n has m rows and n columns. If m = n, a matrix has the same number of rows and columns and is called a square matrix. Matrix A with elements a ij

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 1 Let a.What is the order of A? The matrix has 3 rows and 3 columns, so it is of order 2  3. b.If A = [a ij ], identify a 23 a 32 is in the second row and third column.  3 is in the third row and second column. DNE

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Definition of Equality of Matrices Two matrices A and B are equal if and only if they have the same order m  n and a ij = b ij for i = 1, 2, …, m and j = 1, 2, …, n. For example, if A = B if and only if x = 1 y + 1 = 5 (so y = 4) z = 3 6=6

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 2a Perform the indicated matrix operations:

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Properties of Matrix Addition If A, B, and C are m  n matrices and 0 is the m  n zero matrix, then the following properties are true. 1. A + B = B + A Commutative property of addition 2. (A + B) + C = A + (B + C) Associative property of addition 3. A + 0 = 0 + A = A Additive identity property 4. A + (  A) = (  A) + A = 0 Additive inverse property Definition of a zero matrix: A matrix whose elements are all equal to 0 is called a zero matrix.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 3 If Find  4B. Find 2A + 3B =

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 4 Solve for X in the matrix equation 2X + A = B, where Begin by solving the matrix equation for X. Multiply both sides by ½ rather than divide both sides by 2. This is anticipation of performing scalar multiplication. 2X = B – A X = ½ (B – A )

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 5.1 summary Objectives: Use matrix notation Understand what is meant by equal matrices Add//subtract matrices Perform scalar multiplication Solve matrix equations Vocabulary: Square matrix Equal matrices Zero matrix Additive identity Additive inverse Scalar

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 5.2 Multiplying Matrices and Solving Applications Objectives: Multiply matrices Model applied situations with matrix operations

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Multiply an m  n matrix and an n  p: For the product of two matrices to be defined, the number of columns of the first matrix must equal the number of rows in the second matrix. First MatrixSecond Matrix m  n n  p These must be equal The number of columns in the first matrix must be the same as the number of rows in the second matrix. The order of AB is m  p

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 1a Matrices A and B are defined below. Find AB. Matrix AMatrix B 1  3 3  1 These are equal. The order of AB is 1  1.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 1b Matrices A and B are defined below. Find BA. Matrix BMatrix A 3  1 1  3 These are equal. The order of AB is 3  3.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Matrix multiplication is NOT commutative. AB ≠ BA This was illustrated in the previous two examples.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 2a Where possible find the product. Matrix AMatrix B 2  2 2  4 These are equal. The order is 2  4. 6–

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 2b Where possible find the product. Matrix AMatrix B 2  4 2  2 These are not the same. The number of columns in the first matrix does not equal the number of rows in the second matrix. The product of these two matrices is undefined.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Find AB, given (2x2) (2x2) = 2x2. Example 2c

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall

Example 3 The triangle can be represented by a matrix. Each column in the matrix gives the coordinates of a vertex, or corner. Use matrix operations to perform the following transformations. a. Write the triangle’s vertices as a 2  3 matrix. b. What are the new triangle’s coordinates if the triangle is moved 3 units left and 1 unit down? c. Shrink the triangle’s perimeter by half. continue

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Ex 3 cont. a. Write the triangle’s vertices as a 2  3 matrix. b. What are the new triangle’s coordinates if the triangle is moved 3 units left and 1 unit down? continue

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Ex 3 cont. c. Shrink the triangle’s perimeter by half. Represents the vertices of a triangle with perimeter one-half the original triangle.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 5.2 Summary Objectives: Multiply matrices Model applied situations with matrix operations

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 5.3 Solving Systems of Equations Using Determinants Objectives: Define and evaluate 2x2 determinant Use Cramer’s rule to solve a system of two linear equations in two variables Define and evaluate 3x3 determinant Use Cramer’s rule to solve a system of three equations in three variables

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall If a matrix has the same number of rows and columns, it is called a square matrix. A determinant is a real number associated with a square matrix. The determinant of a square matrix is denoted by placing vertical bars about the array of numbers. Determinant of a 2  2 Matrix

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 1 Evaluate each determinant. a.b. = (-2)(-5) – 3(4) = 10 – 12 = –2 = (3)(-2) – 0(8) = –6 – 0 = –6

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Cramer’s Rule for Two Linear Equations in Two Variables The solution of the system is given by as long as D = ad – bc is not 0.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 2 Use Cramer’s rule to solve the system. First we find D, D x, and D y. The ordered pair solution is (  3, 4).

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 3 Use Cramer’s rule to solve the system. First we find D, D x, and D y. The ordered pair solution is (2, 0).

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Determinant of a 3  3 Matrix

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 4 Evaluate the determinant. Rewrite the first two columns next to the determinant. Multiply along the diagonals.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Cramer’s Rule for Three Equations in Three Variables The solution of the system is given by as long as D is not 0.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 5 Use Cramer’s rule to solve the system. First we find D, D x, D y, and D z = ( ) – (4 – 4 + 6) = 8 = (-4 – 2 + 8) – (-2 – 8 + 4) = 8 = (-4 – 8 – 6) – (8 + 2 – 12) = -16 = (-2 – ) – ( ) = -8 The ordered triple is (1,  2,  1). 4x + 4y – 2z = –2

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 5.3 summary Objectives: By hand: Define and evaluate 2x2 determinant by hand Use Cramer’s rule to solve a system of two linear equations in two variables With calculator: Define and evaluate 3x3 determinant Use Cramer’s rule to solve a system of three equations in three variables - change the directions for the Solve x,y,&z problems to to solve with the calculator Vocabulary: Cramer’s rule Determinant

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 5.4 Multiplicative Inverses and Matrices Objectives: Find the multiplicative inverse of a square matrix

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall For real numbers, we know that 1 is the multiplicative identity because a  1 = 1  a = a. Is there a matrix I such that AI = A and IA = A? A square matrix with 1s down the main diagonal from upper left to lower right and zeros elsewhere does not change the elements in a matrix in products with that matrix.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall The n  n matrix with 1s down the main diagonal from upper left to lower right and 0s elsewhere is called the multiplicative identity matrix of order n. Definition of the Multiplicative Inverse of a Square Matrix Let A be an n  n matrix. If there exists an n  n matrix A  1 (read: “A inverse”) such that then A  1 is the multiplicative inverse of A.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 1 Show that B is the multiplicative inverse of A, where To show that B is the multiplicative inverse of A, we must find the products of AB and BA. Thus, B is the multiplicative inverse of A and we can designate B as

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall If a square matrix has a multiplicative inverse, it is said to be invertible or nonsingular. If a square matrix has no multiplicative inverse, it is called singular. Multiplicative Inverse of a 2  2 Matrix If The matrix A is invertible if and only if ad – bc ≠ 0. If ad – bc = 0 then A does not have a multiplicative inverse.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 2 Find the multiplicative inverse of

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 3 Find the multiplicative inverse of Use a graphing utility to find the inverse matrix. Enter the elements in matrix A and press x  1 to display A  1.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 5.4 summary Objectives: Vocabulary: Multiplicative inverse Nonsingular (inverse exist) Singular (no inverse exist) Find Inverse of 2x2 Matrix by hand Find Inverse of 3x3 Matrix with calculator

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 5.5 Matrix Equations Objectives: Write and solve matrix equations

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Matrix multiplication can be used to represent a system of linear equations. The matrix equation is abbreviated as AX = B, where A is the coefficient matrix of the system. The matrix B is called the constant matrix. Solving a System Using A  1 If AX = B has a unique solution, then X = A  1 B. To solve a linear system of equations, multiply A  1 and B to find X.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Ex 1 Write the system equation from the matrix below 2x + 3y – z = 14 -2x + 4y + z = -20 5y + 6z = 7

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Ex 2 Solve the system of equations using inverse matrices 2x + 3y = -5 5x – 2y = 16

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 3 Solve the system by using A  1, the inverse of the coefficient matrix. Write the linear system. Type in A -1 B and get 2 (0.1, -0.1, 1.8) = ( 1 / 10, - 1 / 10, 9 / 5 )

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 5.5 summary Objectives: Write and solve matrix equations Vocabulary: Coefficient matrix Constant matrix