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Unformatted text preview: 4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1 . A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows: A = a 11 a 12 a 1 n a 21 a 22 a 2 n . . . . . . . . . . . . a m 1 a m 2 a mn = [ a ij ] Definition 4.1.2 . Matrices A and B are equal , A = B , if A and B have the same dimensions and each entry of A is equal to the corresponding entry of B . Discussion Matrices have many applications in discrete mathematics. You have probably encountered them in a precalculus course. We present the basic definitions associated with matrices and matrix operations here as well as a few additional operations with which you might not be familiar. We often use capital letters to represent matrices and enclose the array of numbers with brackets or parenthesis; e.g., A = a b c d or A = a b c d . We do not use simply straight lines in place of brackets when writing matrices because the notation a b c d has a special meaning in linear algebra. A = [ a ij ] is a shorthand notation often used when one wishes to specify how the elements are to be represented, where the first subscript i denotes the row number and the subscript j denotes the column number of the entry a ij . Thus, if one writes a 34 , one is referring to the element in the 3rd row and 4th column. This notation, however, does not indicate the dimensions of the matrix. Using this notation, we can say that two m n matrices A = [ a ij ] and B = [ b ij ] are equal if and only if a ij = b ij for all i and j . Example 4.1.1 . The following matrix is a 1 3 matrix with a 11 = 2 , a 12 = 3 , and a 13 =- 2 . h 2 3- 2 i 4. MATRICES 171 Example 4.1.2 . The following matrix is a 2 3 matrix. - 2 2 5 4.2. Matrix Arithmetic. Let be a scalar, A = [ a ij ] and B = [ b ij ] be m n matrices, and C = [ c ij ] a n p matrix. (1) Addition: A + B = [ a ij + b ij ] (2) Subtraction: A- B = [ a ij- b ij ] (3) Scalar Multiplication: A = [ a ij ] (4) Matrix Multiplication: AC = " n X k =1 a ik c kj # Discussion Matrices may be added, subtracted, and multiplied, provided their dimensions...
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This note was uploaded on 11/27/2011 for the course MCH 108 taught by Professor Penelopekirby during the Fall '08 term at FSU.
- Fall '08