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Linear Algebra

Exercise 2.3.18

Answers

Let $A$ , $B$ , and $C$ be $m \times n$ , $n \times p$ , and $p \times q$ matrices respectively. Next we want to claim that $(AB) C = A (BC)$ since

\begin{array}{l} {((AB) C)}_{ij} & = \sum_{k = 1}^{p} {(AB)}_{ik} C_{kj} = \sum_{k = 1}^{p} (\sum_{l = 1}^{n} A_{il} B_{lk}) C_{kj} \\ = \sum_{k = 1}^{p} \sum_{l = 1}^{n} A_{il} B_{lk} C_{kj} = \sum_{l = 1}^{n} \sum_{k = 1}^{p} A_{il} B_{lk} C_{kj} \\ = \sum_{l = 1}^{n} A_{il} (\sum_{k = 1}^{p} B_{lk} C_{kj}) = \sum_{l = 1}^{n} A_{il} {(BC)}_{lj} = {(A (BC))}_{ij} . \end{array}

jephianlin

2011-06-27 00:00

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