Prove that the set of $n \times n$ permutation matrices is subgroup of $n \times n$ real orthogonal matrices.
I am not sure how to show this permutation matrices are one by switching the rows and columns of $I_n$ but how would you prove this.
2026-03-25 12:49:40.1774442980
Prove that the set of $n \times n$ permutation matrices is subgroup of $n \times n$ real orthogonal matrices.
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First of all, you should realize that permutation matrices are a subset of real orthogonal matrices because the inner product of every two columns (or rows) is zero. In particular, they are invertible. Secondly, you need to show that the product of two permutation matrices is a permutation matrix, and in fact, it corresponds to the composition of corresponding permutations. And finally, check that the inverse is just the matrix obtained by the inverse permutation.
It might be helpful to see how all this works for small $n$, say, $n=3$.