Find a 4 times 4 permutation matrix P such that P4 notequalt

Find a 4 times 4 permutation matrix P such that P^4 notequalto I. For what k does P^k = I?

Solution

For solving the problem, let’s imagine 4 things we are going to permute. Then, we will have to repeat that operation.

As the first step, let’s swap two elements. But we can see that if we repeat that, then we get back to where we started. So P^2 = I which means P^4 = I.

Again, now, think of cyclic permutations: 1, 2, 3, 4 -> 2, 3, 4, 1 -> 3, 4, 1, 2, etc. That has a cycle of 4, so that will not actually work.

But let’s think of cyclic permutations of 3 variables. 1, 2, 3, 4 -> 2, 3, 1, 4 -> 3, 1, 2, 4, etc. That has a period of 3. Thus, after the 4th permutation, we definitely aren\'t back to where we started.

Thus, following by the last method, let’s write the permutation that rotates among the first 3 elements and leaves the 4th alone, as a permutation matrix. Thusm, we can recah our answer.