Download Matrix Arithmetic, Addition and Subtraction of Matrices | MATH 1630 and more Study notes Mathematics in PDF only on Docsity! Matrix Arithmetic ~ page 1 J. Ahrens ~ 1/9/2006 MATRIX ARITHMETIC Addition and Subtraction of Matrices 1. A matrix of order m x n has m rows and n columns [row x column] a. In a square matrix m = n b. In a row matrix m = 1 c. In a column matrix n = 1 2. Two matrices are equal if they have the same order and each pair of corresponding elements is equal. a. Given that: 9 7 m 3 n 5 r 0 8 0 − + = b. Then: 9 = m - 3 | m = 12 7 = n + 5 | n = 2 r = 8 3. To add/subtract two matrices of the same order, add/subtract the corresponding elements. You cannot add or subtract matrices if they are not the same size. 4. The additive inverse (negative) of matrix X is formed by changing each element to its opposite and is denoted by -X. 5. The additive identity (zero matrix) has zeros for all elements. The zero matrix is not unique; their is one to match a matrix of any size. 6. Examples: Given that A = and B = 4 3 0 5 1 0 3 7 − − 1 4 2 6 1 8 3 5 − − − A + B = 5 7 2 11 0 8 0 12 − − A – B = 3 1 2 1 2 8 6 2 − − − - A = 4 3 0 5 1 0 3 7 − − − − The 0 matrix of order 2 x 4 is 0 0 0 0 0 0 0 0 4 3 1 1 1 1 4 3 1 2 1 0 1 4 1 6 − + = Matrix Arithmetic ~ page 2 J. Ahrens ~ 1/9/2006 Scalar Multiplication 1. A scalar is a real number, not a matrix 2. To multiply a scalar times a matrix, multiply each element of the matrix by the scalar. If A = , then 3A = 3 = 4 3 0 5 1 0 3 7 − − 4 3 0 5 1 0 3 7 − − 12 9 0 15 3 0 9 21 − − 3. If B = , then 3A - 4B = - 4− − − 1 4 2 6 1 8 3 5 − − 12 9 0 15 3 0 9 21 − − − 1 4 2 6 1 8 3 5 = - − − 12 9 0 15 3 0 9 21 − − − 4 16 8 24 4 32 12 20 = − − − 8 7 8 9 7 32 21 1 Matrix Multiplication 1. True matrix multiplication can be very tricky. a. It is not always possible to multiple two given matrices, even if they are the same size. b. It may be possible to multiply A times B (in that order), but not B times A. c. It may be possible to multiply A times B and B times A, but not get the same answer! 2. The sizes of the matrices will determine whether or not they can be multiplied. a. Suppose that we have matrix A which is 3x2 and matrix B which is 2x3. b. Write the orders of the matrices side-by-side and compare the two “inner” numbers c. If the number of columns in the first matrix is equal to the number of rows in the second matrix, then the product AB can be found. d. If the product exists, then its size is given by the two “outer” numbers. = ↑ ↑ matrix matrix 3 x 2 2 x 3 must be will be 3 x 3 A B AB e. Would we be able to find BA in this case? Yes, it will be a 2x2 matrix. = ↑ ↑ matrix matrix 2 x 3 3 x 2 must be will be 2 x 2 B A BA 3. Matrices form a mathematical system of their own, so they do not have to follow the same rules that the “real” numbers do.
