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Unformatted text preview: Chapter 1. Matrices, determinants, system of linear equa- tions 1.1. 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 1.1 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 ) if the size of A is understood. Denote by M m × n ( R ) the set of all m × n matrices with real entries. 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 . Denote by M n ( R ) the set of all square matrices of order n with real entries. 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 is a 4 × 1 matrix (column vector). Example 1.2 A zero matrix m × n ∈ M m × n ( R ) (or just 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 . Another special matrix is the n × n identity matrix whose entries δ ij are the Kronecker delta : δ ij = 1 if i = j ; if i 6 = j, for 1 ≤ i, j ≤ n , i.e. the diagonal entries are all one while the off-diagonal entries are all zero. Often the identity matrices are denoted by I n . As you will see later, the zero and identity matrices play a similar role as 0 and 1 in the arithmetic of real numbers. 2 Definition 1.3 If A, B ∈ M m × n ( R ) then we say that A = B provided corresponding 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 1.4 Let A, B ∈ M m × n ( R ) , with A = ( a ij ) and B = ( b ij ) . 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. Next we proceed to multiplication involving matrices. Note that we can define two kinds of multiplication, namely scalar multiplication and matrix multiplication. We first look at scalar multiplication: Definition 1.5 Let A = ( a ij ) m × n ∈ M m × n ( R ) . For any λ ∈ R , the scalar multiple of A by λ is defined by λA = ( λa ij ) m × n . In particular,....
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