Chapter2 - Chapter 2 Matrices Section 2.1 Introduction to...

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Unformatted text preview: Chapter 2 Matrices Section 2.1 Introduction to Matrices Definition 2.1.1 A matrix (plural matrices ) is a rectangular array of numbers. The numbers in the array are called entries in the matrix. The size of a matrix is given by m × n where m is the number of rows and n is the number of columns. The ( i,j )- entry of a matrix is the number which is in the i th row and j th column of the matrix. Example 2.1.2 1.   1 2 3 4- 1   is a 3 × 2 matrix. The (1,2)-entry of the matrix is 2 and the (3,1)-entry is 0. 2. ( 2 1 0 ) is a 1 × 3 matrix. 3.   √ 2 3 . 1- 2 3 1 2 π   is a 3 × 3 matrix. 4. ( 4 ) is a 1 × 1 matrix. 5.   1 1 2   is a 3 × 1 matrix. Definition 2.1.3 A column matrix (or a column vector ) is a matrix with only one column. A row matrix (or a row vector ) is a matrix with only one row. 2 Chapter 2. Matrices Example 2.1.4 The matrix in Example 2.1.2.5 is a column matrix and the matrix in Example 2.1.2.2 is a row matrix. The matrix in Example 2.1.2.4 is both a column and row matrix. Notation 2.1.5 In general, an m × n matrix can be written as A =      a 11 a 12 ··· a 1 n a 21 a 22 ··· a 2 n . . . . . . . . . a m 1 a m 2 ··· a mn      . or simply A = ( a ij ) m × n where a ij is the ( i,j )-entry of A . Sometimes, if the size of the matrix is already known, we may just write A = ( a ij ). Example 2.1.6 Write down the following matrices explicitly: 1. A = ( a ij ) 2 × 3 where a ij = i + j . 2. B = ( b ij ) 3 × 2 where b ij = ‰ 1 if i + j is even- 1 if i + j is odd. Answers 1. A = 2 3 4 3 4 5 ¶ ; 2. B =   1- 1- 1 1 1- 1   . Definition 2.1.7 The following are some special types of matrices: 1. A matrix is called a square matrix if it has the same number of rows and columns. In particular, an n × n square matrix is called a square matrix of order n . 2. Given a square matrix A = ( a ij ) of order n , the diagonal of A is the sequence of entries a 11 ,a 22 ,...,a nn . The entries a ii are called the diagonal entries while a ij , i 6 = j , are called non-diagonal entries . A square matrix is called a diagonal matrix if all its non-diagonal entries are zero, i.e. A = ( a ij ) n × n is diagonal ⇔ a ij = 0 whenever i 6 = j. 3. A diagonal matrix is called a scalar matrix if all its diagonal entries are the same, i.e. A = ( a ij ) n × n is scalar ⇔ a ij = ‰ 0 if i 6 = j c if i = j for a constant c . Section 2.1. Introduction to Matrices 3 4. A diagonal matrix is called an identity matrix if all its diagonal entries are 1. We use I n to denote the identity matrix of order n . Sometimes we write I instead of I n when there is no danger of confusion. 5. A matrix with all entries equal to zero is called a zero matrix . We denote the m × n zero matrix by m × n , or simply ....
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This note was uploaded on 01/24/2011 for the course SPMS MAS 114 taught by Professor Profdavidh.adams during the Spring '10 term at Nanyang Technological University.

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Chapter2 - Chapter 2 Matrices Section 2.1 Introduction to...

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