796 CHAPTER 11 Systems of Equations and Inequalities The matrix developed in Example 1 has 2 rows and 3 columns. In general, a matrix with m rows and n columns is called an m by n matrix . An m by n matrix contains m n ⋅ entries. The matrix developed in Example 1 is a 2 by 3 matrix and contains 2 3 6 ⋅ = entries. If an m by n matrix has the same number of rows as columns, that is, if m n, = then the matrix is a square matrix . We can arrange these data in a rectangular array as follows: Too High Too Low No Opinion Man 200 150 45 Woman 315 125 65 or as the matrix 200 150 45 315 125 65 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ This matrix has two rows (representing man and woman) and three columns (representing “too high,” “too low,” and “no opinion”). In Words For an m by n matrix, the number of rows is listed first and the number of columns second. Examples of Matrices (a) − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 5 0 6 1 A 2 by 2 square matrix (b) 103 ⎡ ⎣⎢ ⎤ ⎦⎥ A 1 by 3 matrix (c) 6 2 4 4 3 5 8 0 1 ⎡ − ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ A 3 by 3 square matrix EXAMPLE 2 NOTE Matrices and determinants are different. A matrix is a rectangular array of numbers or expressions, nothing more. A determinant has a value, usually a real number. j 1 Find the Sum and Difference of Two Matrices We begin our discussion of matrix algebra by defining equal matrices and then defining the operations of addition and subtraction. It is important to note that these definitions require both matrices to have the same number of rows and the same number of columns as a condition for equality and for addition and subtraction. Matrices usually are represented by capital letters, such as A, B, and C. DEFINITION Equal Matrices Two matrices A and B are equal , written as A B = provided that A and B have the same number of rows and the same number of columns and each entry aij in A is equal to the corresponding entry bij in B. For example, 2 1 0.5 1 4 1 1 2 1 and 3 2 1 0 1 2 9 4 1 0 1 8 4 1 6 1 4 0 6 1 4 1 2 6 1 2 4 1 2 3 6 1 2 4 3 − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ≠ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ≠ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ Because the entries in row 1, column 2 are not equal; that is, ≠ a b 12 12 Because the matrix on the left has 3 columns and the matrix on the right has 4 columns Suppose that A and B represent two m by n matrices. The sum , A B, + is defined as the m by n matrix formed by adding the corresponding entries aij of A and bij of B. The difference , A B, − is defined as the m by n matrix formed by subtracting the entries bij in B from the corresponding entries aij in A. Addition and subtraction of matrices are defined only for matrices having the same number m of rows and the same number n of columns. For example, a 2 by 3 matrix and a 2 by 4 matrix cannot be added or subtracted.
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