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Unformatted text preview: 22 Chapter 2 MATRICES 2.1 Matrix arithmetic A matrix over a field F is a rectangular array of elements from F . The sym bol M m × n ( F ) denotes the collection of all m × n matrices over F . Matrices will usually be denoted by capital letters and the equation A = [ a ij ] means that the element in the i –th row and j –th column of the matrix A equals a ij . It is also occasionally convenient to write a ij = ( A ) ij . For the present, all matrices will have rational entries, unless otherwise stated. EXAMPLE 2.1.1 The formula a ij = 1 / ( i + j ) for 1 ≤ i ≤ 3 , 1 ≤ j ≤ 4 defines a 3 × 4 matrix A = [ a ij ], namely A = 1 2 1 3 1 4 1 5 1 3 1 4 1 5 1 6 1 4 1 5 1 6 1 7 . DEFINITION 2.1.1 (Equality of matrices) Matrices A and B are said to be equal if A and B have the same size and corresponding elements are equal; that is A and B ∈ M m × n ( F ) and A = [ a ij ] , B = [ b ij ], with a ij = b ij for 1 ≤ i ≤ m, 1 ≤ j ≤ n . DEFINITION 2.1.2 (Addition of matrices) Let A = [ a ij ] and B = [ b ij ] be of the same size. Then A + B is the matrix obtained by adding corresponding elements of A and B ; that is A + B = [ a ij ] + [ b ij ] = [ a ij + b ij ] . 23 24 CHAPTER 2. MATRICES DEFINITION 2.1.3 (Scalar multiple of a matrix) Let A = [ a ij ] and t ∈ F (that is t is a scalar ). Then tA is the matrix obtained by multiplying all elements of A by t ; that is tA = t [ a ij ] = [ ta ij ] . DEFINITION 2.1.4 (Additive inverse of a matrix) Let A = [ a ij ] . Then A is the matrix obtained by replacing the elements of A by their additive inverses; that is A = [ a ij ] = [ a ij ] . DEFINITION 2.1.5 (Subtraction of matrices) Matrix subtraction is defined for two matrices A = [ a ij ] and B = [ b ij ] of the same size, in the usual way; that is A B = [ a ij ] [ b ij ] = [ a ij b ij ] . DEFINITION 2.1.6 (The zero matrix) For each m, n the matrix in M m × n ( F ), all of whose elements are zero, is called the zero matrix (of size m × n ) and is denoted by the symbol 0. The matrix operations of addition, scalar multiplication, additive inverse and subtraction satisfy the usual laws of arithmetic. (In what follows, s and t will be arbitrary scalars and A, B, C are matrices of the same size.) 1. ( A + B ) + C = A + ( B + C ); 2. A + B = B + A ; 3. 0 + A = A ; 4. A + ( A ) = 0; 5. ( s + t ) A = sA + tA , ( s t ) A = sA tA ; 6. t ( A + B ) = tA + tB , t ( A B ) = tA tB ; 7. s ( tA ) = ( st ) A ; 8. 1 A = A , 0 A = 0, ( 1) A = A ; 9. tA = 0 ⇒ t = 0 or A = 0. Other similar properties will be used when needed. 2.1. MATRIX ARITHMETIC 25 DEFINITION 2.1.7 (Matrix product) Let A = [ a ij ] be a matrix of size m × n and B = [ b jk ] be a matrix of size n × p ; (that is the number of columns of A equals the number of rows of B ). Then AB is the m × p matrix C = [ c ik ] whose ( i, k )–th element is defined by the formula c ik = n X j =1 a ij b jk = a i 1 b 1 k + ··· + a in b nk ....
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This note was uploaded on 04/23/2008 for the course MA 265 taught by Professor Bens during the Spring '08 term at Purdue.
 Spring '08
 Bens
 Linear Algebra, Algebra, Matrices

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