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2. Matrix Algebra 2.1 Matrix Operations j-th column A ai , j a11 ai1 am1 a1 j aij amj a1n ain amn i-th row Diagonal entries Diagonal matrix : a square matrix whose nondiagonal entries are zero. Recall: Two matrices are equal the matrices are the same size and their corresponding entries are equal. Theorem 1 Let A, B, and C be matrices of the same size, and let r and s be scalars. A B B A ( A B) C A ( B C ) r ( A B ) rA rB (r s ) A rA sA A0 A r ( sA) ( rs ) A Example: 1 0 2 A : 3 1 0 4 2 1 2 1 1 B : 2 1 0 3 0 2 A B A B 2B 1 A 3B 2 A2 Example: 1 A := 4 -7 -2 -5 8 -3 6 9 1 -7 C := 4 0 -3 9 -6 0 5 -2 8 0 AC AC 2C 1 A 3C 2 Matrix Multiplication REVIEW x1 x 2 Ax a1 a 2 L a n x1a1 x 2a 2 L x n a n M x n Recall: 1 1 3 2 1 1 3 1 Ax 2 0 2 1 2210 0 2 4 3 2 2 0 3 2 2 8 Matrix Multiplication If A is an m n matrix and B is an n p matrix with columns b1,b2 ,L ,b p , then the product AB is the m p matrix whose columns are A b1,K ,Ab p AB A b1 b2 L b p A b1 A b2 L A b p Example: Let Find AB 1 2 A 0 1 4 5 2 3 B 1 2 Row-Column Rule for Computing AB: If the product AB is defined, then the entry in row i and column j of AB is the sum of the products of corresponding entries from row i of A and column j of B. ( AB) ij ai1b1 j ai 2b2 j aipb pj ainbnj Example: 1 2 _______ _______ 0 1 2 3 _______ _______ 1 2 4 2 5 1 _______ 4 5 3x2 2x2 3x2 Properties of Matrix Multiplication A( BC ) ( AB)C A( B C ) AB AC ( B C ) A BA CA r ( AB) (rA) B A(rB ) for any scalar r I m A A AI n for m n matrix A In general the followings are NOT true. AB BA If AB AC then B C If AB 0 then A 0 or B 0 Defn: Given an m×n matrix A, the transpose of A is the n×m matrix, denoted by AT, whose columns are formed from the corresponding rows of A. A (aij ) AT (a ji ) 3 1 0 Example: Let A : 1 2 4 What is AT ? Rules related to transpose: (A ) A T T ( A B) A B T T T (rA)T rAT for any scalar r ( AB) B A T T T