I'm working on an assignment for an intro to relativity class and I've come up against this series of equations.

[![enter image description here](https://i.stack.imgur.com/4bwlY.png)](https://i.stack.imgur.com/4bwlY.png)

The step I'm confused about is going from summation notation to matrix notation where they've taken the transpose of one of the Lorentz transform matrices. I understand that you *can* do this, but I don't understand *why*. What's the motivation? Wouldn't the result (det(L)=+-1) be exactly the same if you didn't take the transpose? There's obviously something I'm missing here, but I don't know what.

EDIT: 

After reading a few responses I'm editing this to add that what I'm asking is if someone can explain to me (in detail, showing all the steps) how to go from tensor notation to matrix notation. I think (I hope) that will make it clear what's happening between eqns. 1.7 and 1.10.