Problem Goldstein 4-1 3 rd ed 41 Associativity Say we have...

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Problem Goldstein 4-1 (3 rd ed. 4.1) Associativity Say we have three matrices, P , Q , R , whose dimensions are such that the matrix multiplications PQ = S , (1) and QR = T , (2) are de fi ned. Then, to prove associativity, we need to show that SR = PT . (3) Consider the ij element of each side of Eq.(3), ( SR ) ij = X k S ik R kj , (4) ( PT ) ij = X k P ik T kj . (5) From Eqs.(1) and (2), we have S ik = ( PQ ) ik = X l P il Q lk , (6) and T kj = ( QR ) kj = X l Q kl R lj . (7) Next we substitute Eq.(6) into Eq.(4) to obtain ( SR ) ij = X k X l P il Q lk R kj , (8) and we substitute Eq.(7) into Eq.(5) to obtain ( PT ) ij = X k P ik X l Q kl R lj = X k X l P ik Q kl R lj . (9) In Eq.(8), we can invert the order of summation to get ( SR ) ij = X l X k P il Q lk R kj , (10) 1