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1/6/2015 What You Should Learn • Decide whether two matrices are equal. 8.2 OPERATIONS WITH MATRICES • Add and subtract matrices and multiply matrices by scalars. • Multiply two matrices. • Use matrix operations to model and solve real-life problems. Copyright © Cengage Learning. All rights reserved. 2 Equality of Matrices Equality of Matrices Two matrices A = [aij] and B = [bij] are equal if they have There is a rich mathematical theory of matrices, and its applications are numerous. This section introduces some fundamentals of matrix theory. the same order (m ´ n) and aij = bij for 1 £ i £ m and 1 £ j £ n. It is standard mathematical convention to represent matrices in any of the following three ways. In other words, two matrices are equal if their corresponding entries are equal. 3 Example 1 – Equality of Matrices 4 Matrix Addition and Scalar Multiplication Solve for a11, a12, a21, and a22 in the following matrix equation. In this section, three basic matrix operations will be covered. The first two are matrix addition and scalar multiplication. With matrix addition, you can add two matrices (of the same order) by adding their corresponding entries. 5 6 1 1/6/2015 Example 2 – Addition of Matrices Example 2 – Addition of Matrices cont’d d. The sum of and is ___________ because A is of order 3 ´ 3 and B is of order 3 ´ 2. 7 Matrix Addition and Scalar Multiplication 8 Matrix Addition and Scalar Multiplication The symbol –A represents the negation of A, which is the scalar product (–1)A. In operations with matrices, numbers are usually referred to as scalars. In this text, scalars will always be real numbers. Moreover, if A and B are of the same order, then A – B represents the sum of A and (–1)B. You can multiply a matrix A by a scalar c by multiplying each entry in A by c. That is, A – B = A + (–1)B. Subtraction of matrices The order of operations for matrix expressions is similar to that for real numbers. 9 Example 3 é4 If A = ê 0 ê êë- 3 - 1ù 4ú ú 8 úû 10 Matrix Addition and Scalar Multiplication é 0 4ù and B = ê- 1 3ú find: ê ú êë 1 7úû The properties of matrix addition and scalar multiplication are similar to those of addition and multiplication of real numbers. a. -3A b. 2B c. -3A+2B 11 12 2 1/6/2015 Matrix Addition and Scalar Multiplication Matrix Addition and Scalar Multiplication Note that the Associative Property of Matrix Addition allows you to write expressions such as A + B + C without ambiguity because the same sum occurs no matter how the matrices are grouped. One important property of addition of real numbers is that the number 0 is the additive identity. That is, c + 0 = c for any real number c. For matrices, a similar property holds. This same reasoning applies to sums of four or more matrices. That is, if A is an m ´ n matrix and O is the m ´ n zero matrix consisting entirely of zeros, then A + O = A. 13 14 Matrix Addition and Scalar Multiplication Matrix Addition and Scalar Multiplication In other words, O is the additive identity for the set of all m ´ n matrices. The algebra of real numbers and the algebra of matrices have many similarities. For example, the following matrices are the additive identities for the sets of all 2 ´ 3 and 2 ´ 2 matrices. For example, compare the following solutions. 2 ´ 3 zero matrix m ´ n Matrices (Solve for X.) Real Numbers (Solve for x.) and 2 ´ 2 zero matrix x + a= b X+A=B x + a + (–a) = b + (–a) X + A + (–A) = B + (–A) x+0 =b–a X+O=B–A x=b–a X=B–A 15 Example 6 Matrix Multiplication Another basic matrix operation is matrix multiplication. Solve for X in the equation 2X-2A=B, where A = é 0 3 ù and B = é- 4 - 8ù ê2 0ú ê ú ëê- 4 - 1ûú 16 ê- 2 0 ú ê ú ëê 14 - 6ûú The definition of matrix multiplication indicates a row-by-column multiplication, where the entry in the i th row and jth column of the product AB is obtained by multiplying the entries in the i th row of A by the corresponding entries in the jth column of B and then adding the results. 17 18 3 1/6/2015 Example 7 – Finding the Product of Two Matrices Matrix Multiplication So for the product of two matrices to be defined, the number of columns of the first matrix must equal the number of rows of the second matrix. Find the product AB using and The general pattern for matrix multiplication is as follows. 19 Matrix Multiplication 20 Example 9 Be sure you understand that for the product of two matrices to be defined, the number of columns of the first matrix must equal the number of rows of the second matrix. Find the product and dimension of each of the following: a. é 0 1 ù é 0 3 - 2ù That is, the middle two indices must be the same. The outside two indices give the order of the product, as shown below. A ´ B = AB b. ê- 4 m´n n´p ê7 2ú ê 2 1 ë ûë é 2 ê ëê 1 m´p c. 4 úû 1ù é3ù 0úú ê ú -1 3úû ë û é 0 2 4ù é 2 - 4 5ù ê- 1 1 2 ú ê 0 9 3 ú ë ûë û 21 Matrix Multiplication 22 Matrix Multiplication Even if both AB and BA are defined, matrix multiplication is not, in general, commutative. That is, for most matrices, AB ¹ BA. This is one way in which the algebra of real numbers and the algebra of matrices differ. 23 24 4 1/6/2015 Matrix Multiplication Applications If A is an n ´ n matrix, the identity matrix has the property that AIn = A and In A = A. For example, Matrix multiplication can be used to represent a system of linear equations. AI = A Note how the system can be written as the matrix equation AX = B, where A is the coefficient matrix of the system, and X and B are column matrices. and IA = A A 25 ´ X = B 26 Example 11 – Solving a System of Linear Equations Consider the following system of linear equations. x1 – 2x2 + x3 = – 4 x2 + 2x3 = 4 2x1 + 3x2 – 2x3 = 2 a. Write this system as a matrix equation, AX = B. b. Use Gauss-Jordan elimination on the augmented matrix to solve for the matrix X. 27 5