Question

What do the hats in Problem 5.1 (2.7 #13) mean?
"Find a 4 by 4 permutation Phatwith Phat^4= I"

Answer 1

I don't know what the problem is (I'm not taking the course), but hats generally mean that a vector is of magnitude 1 (sometimes it can just mean basis vector).

Answer 2

I just read the question, and I think the hats are to distinguish it from the permutation matrix in the first half of the question. Strang seems to do that occaisionally when he wants to stress that one matrix is similar to, but different from another matrix. Later, in the section on Least squares, 4.3, he takes the equation Ax = b, and says that when it doesn't have a solution, we can find the best solution \[A\hat x = p \] His point is that x-hat is different from x, but similar in that it is the best solution to an unsolvable problem.

In this case, I'm fairly certain he wants any 4 by 4 permutation matrix where \[\hat P^4 \ne I \]
As an example, here is a 3 by 3 matrix which when squared, does not equal I.\[\left[\begin{matrix}0 & 1&0 \\ 0 & 0&1\\1&0&0\end{matrix}\right]\]