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Unformatted text preview: McGill University Math 270: Applied Linear Algebra CHAPTER 2: MATRICES AND DETERMINANTS 1 Matrix Algebra Suppose given two matrices A = [ a ij ] and B = [ b ij ] . The two matrices are equal, A = B, if and only if • They have the same dimensions; • All corresponding elements are equal: a ij = b ij for all ( i,j ) . 1.1 Matrix Addition Addition (or subtraction) of A and B with the same dimension gives the matrix C = [ c ij ] = A ± B : c ij = ( A ± B ) ij = a ij ± b ij for all ( i,j ) . 00 1.2 Scalar Multiplication of Matrix Multiplication of a matrix by a scalar, C = [ c ij ] = αA : c ij = ( αA ) ij = αa ij for all ( i,j ) . 1.3 Matrix Multiplication Given A = [ a ij ] m × n , B = [ b ij ] n × p . The multiplication of A and B is defined as C = [ c ij ] n × p = AB : c ij = ( AB ) ij = n ∑ k =1 a ik b kj ( i = 1 , ··· ,m ); ( j = 1 , ··· ,p ) . From the above, we derive 1. Given matrix A = [ a ] m × n , and a column n − vector b = [ b i ] n , which implies p = 1 . then we have ( A b ) i = n ∑ k =1 a ik b k ( i = 1 , ··· ,m ); 2. Given matrix A = [ a ] m × n , and B = [ b 1 , b 2 , ··· , b p ] , then we have AB = [ A b 1 ,A b 2 , ··· A b p ] 01 1.4 Some Properties of Matrix Multiplication Suppose that matrices A,B,C have appropriate dimensions for the operations to be performed. Then, one has: • Matrix multiplication is associative and distributive over addition: A ( BC ) = ( AB ) C A ( B + C ) = AB + AC ( A + B ) C = AC + BC • Matrix multiplication is not commutative: AB ̸ = BA. 2 Some Important ( n × n ) Matrices Suppose we have the matrix, A = [ a ij ] n × n . • A is the zero matrix , if a ij = 0 , for i = 1 , ··· ,n ; j = 1 , ··· ,n. • A is the identity matrix , A = I n if a ij = δ ij = { 1 , ( i = j ) , ( i ̸ = j ) . It follows that I n I n = I n . 02 • A is the upper (or lower) triangular matrices , if a ij = 0 , for i > j, (or i < j ) . • A is the diagonal matrices , if a ij = 0 , for i ̸ = j. Namely, A = D = d 11 ··· d 22 ··· . . . ··· d nn It is derived D k = d k 11 ··· d k 22 ··· . . . ··· d k nn • A is the symmetric matrices , if A is square ( m = n ) , and a ij = a ji . • A is the antisymmetric matrices or skew symmetric , if A is square ( m = n ) , and a ij = − a ji . 03 2.1 Row Elementary Operation Matrices Recall that in last chapter, we have defined three types of ele mentary row operations. By allying these operations, one that transform the ( n × n ) matrix ( A ) into ( ˜ A ) . Specifically, • By (EOP)1: R i ⇐⇒ R j , ( A ) ⇐⇒ ( ˜ A ) 1 ; • By (EOP)2: R i ⇐⇒ kR i , ( A ) ⇐⇒ ( ˜ A ) 2 ; • By (EOP)3: R i ⇐⇒ R i + kR j , ( A ) ⇐⇒ ( ˜ A ) 3 ....
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 Winter '07
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 Determinant, Matrices, Det

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