University:
Stanford University
Course:
MATH 113 | Linear Algebra and Matrix Theory
Academic year:

2022
Views:

65

Pages:

4
Author:

Carissa Thomas

1.3 MATRICES AND MATRIX OPERATIONS Rectangular arrays of real numbers arise in many contexts other than as augmented matrices for systems of linear equations. In this section we begin our study of matrix theory by giving some of the fundamental definitions of the subject. We shall see how matrices can be combined through the arithmetic operations of addition, subtraction, and multiplication. Matrix Notation and Terminology In Section 1.2 we used rectangular arrays of numbers, called augmented matrices, to abbreviate systems of linear equations. However, rectangular arrays of numbers occur in other contexts as well. For example, the following rectangular array with three rows and seven columns might describe the number of hours that a student spent studying three subjects during a certain week: Mon. Tues. Wed. Thurs. Fri. Sat. Sun. Math 2 3 2 4 1 4 2 History 0 3 1 4 3 2 2 Language 4 1 3 1 0 0 2 If we suppress the headings, then we are left with the following rectangular array of numbers with three rows and seven columns, called a "matrix": [ 2 3 2 4 1 4 2 ] [ 0 3 1 4 3 2 2 ] [ 4 1 3 1 0 0 2 ] DEFINITION A matrix is a rectangular array of numbers. The numbers in the array are called the entries in the matrix. EXAMPLE 1 Examples of Matrices Some examples of matrices are [ 1 2 ] [ 3 0 ] [ -1 4 ] [ 2 1 0 -3 ], [ e π -√2 ] [ 0 ½ 1 ] [ 0 0 0 ] [ 1/3 ], [ 4 ] The size of a matrix is described in terms of the number of rows (horizontal lines) and columns (vertical lines) it contains. For example, the first matrix in Example 1 has three rows and two columns, so its size is 3 by 2 (written 3 x 2). In a size description, the first number always denotes the number of rows, and the second denotes the number of columns. The remaining matrices in Example 1 have sizes 1 x 4, 3 x 3, 3 x 1, and 1 x 1, respectively. A matrix with only one column is called a column matrix (or a column vector), and a matrix with only one row is called a row matrix (or a row vector). Thus, in Example 1 the 3 x 1 matrix is a column matrix, the 1 x 4 matrix is a row matrix, and the 1 x 1 matrix is both a row matrix and a column matrix. (The term vector has another meaning that we will discuss in subsequent chapters.) Remark It is common practice to omit the brackets from a 1 x 1 matrix. Thus we might write 4 rather than. Although this makes it impossible to tell whether 4 denotes the number "four" or the 1 x 1 matrix whose entry is "four," this rarely causes problems, since it is usually possible to tell which is meant from the context in which the symbol appears. We shall use capital letters to denote matrices and lowercase letters to denote numerical quantities; thus we might write A = [ 2 1 7 ] [ 3 4 2 ] or C = [ a b c ] [ d e f ] A = [ a₁₁ a₁₂ a₁₃ a₁₄ ] [ a₂₁ a₂₂ a₂₃ a₂₄ ] [ a₃₁ a₃₂ a₃₃ a₃₄ ] and a general m x n matrix as A = [ a₁₁ a₁₂ ... a₁n ] [ a₂₁ a₂₂ ... a₂n ] [ ⋮ ⋮ ⋮ ] [ aₘ₁ aₘ₂ ... aₘn ] (1) When compactness of notation is desired, the preceding matrix can be written as the first notation being used when it is important in the discussion to know the size, and the second being used when the size need not be emphasized. Usually, we shall match the letter denoting a matrix with the letter denoting its entries; thus, for a matrix B we would generally use b< sub>ij< /sub> for the entry in row i and column j, and for a matrix C we would use the notation c< sub>ij< /sub>.