The problem asks to establish the following properties of permutation matrices, for all .

1. iff for all , where denotes the identity matrix.

2. .

3. .

4. is an orthogonal matrix, that is, .

Can someone pls check if my attempts below are correct proofs? Does the first one below qualify as a proof?

1. If for all then does not swap any rows of and therefore which implies . If then meaning for all .

2. swaps th row of with th row which is then swapped with th row. So th row is swapped with th row which is what does. Therefore .

3. Property 2 says that . Because for all , property 1 says . Hence and .

4. If , then the th row of will be the th row of . The inverse must copy the th row back to the th row and therefore . Let all other entries of be zero, since all other entries are also zero, we have .