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Chapter7(1804)

# Chapter7(1804) - Ch7/MATH1804/YMC/2009-10 1 Chapter 7...

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Ch7/MATH1804/YMC/2009-10 1 Chapter 7. Matrices, determinants, system of linear equations, eigenvalues and eigenvectors 7.1. Matrix Arithmetic and Operations This section is devoted to developing the arithmetic of matrices. We will see some of the differences between arithmetic of real numbers and matrices. Definition 7.1 Let m , n be positive integers. An m × n matrix A is an array of real numbers A = a 11 a 12 · · · a 1 n a 21 a 22 · · · a 2 n . . . . . . . . . . . . a m 1 a m 2 · · · a mn where a ij R is the ( i, j ) -th entry of A . We shall write A = ( a ij ) 1 i m ;1 j n for short, or A = ( a ij ) m × n , or A = ( a ij ) if the size of A is understood. Matrices of the shape m × 1 are called column vectors , whereas matrices of the shape 1 × n are called row vectors . Note that the integer m need not be equal to n . In the case of m = n , we have a n × n matrix, and it is called a square matrix of order n . The following are some examples of matrices: A = a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 ! , B = 1 9 2 7 5 - 3 - 5 8 4 , C = 4 - 6 5 1 Matrix A is a general 2 × 4 matrix with entries a ij R where 1 i 2 and 1 j 4 . Matrix B is a square matrix of order 3 , and C a 4 × 1 matrix (column vector). Example 7.2 A zero matrix 0 m × n (or just 0 if the size is understood) is a matrix with all its entries equal to zero, i.e. a ij = 0 for 1 i m and 1 j n . Example 7.3 A square matrix of order n is the identity matrix (of order n ) if all the diagonal entries are one while all the off-diagonal entries are zero. We shall denote the identity matrix of order n by I n , or simply I if there is no ambiguity. For example, when n = 2 , 3 , we have: I 2 = 1 0 0 1 ! , I 3 = 1 0 0 0 1 0 0 0 1 As we will see later, the zero and identity matrices play a similar role as 0 and 1 in the arithmetic of real numbers.

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Ch7/MATH1804/YMC/2009-10 2 Definition 7.4 If A and B are both m × n matrices, then we say that A = B provided the corre- sponding entries from each matrix are equal, that is, A = B provided a ij = b ij for all i and j . Matrices of different sizes cannot be equal. Now we define addition and subtraction of matrices: Definition 7.5 Let A = ( a ij ) m × n and B = ( b ij ) m × n . Then the sum and the difference of A and B , written as A + B and A - B , are also m × n matrices with entries given by a ij + b ij and a ij - b ij respectively. Matrices of different sizes cannot be added or subtracted. Example 7.6 1 3 7 2 - 2 3 ! + 0 - 1 5 2 3 1 ! = 1 + 0 3 + ( - 1) 7 + 5 2 + 2 - 2 + 3 3 + 1 ! = 1 2 12 4 1 4 ! . Next we proceed to multiplication involving matrices. Note that we can define two kinds of mul- tiplication, namely scalar multiplication and matrix multiplication. We first look at scalar multiplication: Definition 7.7 Let A = ( a ij ) m × n . For any λ R , the scalar multiple of A by λ is defined by λA = ( λa ij ) m × n . In particular, ( - 1) A is simply written as - A .
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Chapter7(1804) - Ch7/MATH1804/YMC/2009-10 1 Chapter 7...

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