Understanding the Permitted Use of Double Summation in Math

[IniquiTrance]

If I know that \sum_{k=1}^n a_{ik} = 1 and \sum_{j=1}^n b_{kj} = 1, why is the following permitted?

\sum_{j=1}^n \sum_{k=1}^n a_{ik}b_{kj} = \left(\sum_{j=1}^n b_{kj}\right) \left(\sum_{k=1}^n a_{ik}\right) = 1\cdot 1 = 1


Thanks!

 

[micromass]

It's not permitted. What you wrote makes no sense since you sum over k and one of the b_{kj} is outside that sum. That is not allowed.

What you could do is:

\sum_{j=1}^n\sum_{k=1}^n a_{ik}b{kj}= \sum_{k=1}^n \sum_{j=1}^n a_{ik}b_{kj} = \sum_{k=1}^n \left(a_{ik} \sum_{j=1}^n b_{kj}\right)= \sum_{k=1}^n a_{ik}=1

 

[IniquiTrance]

Thank you very much. I knew I wasn't understanding something.