Identity Matrix is Permutation Matrix

Theorem
An identity matrix is an example of a permutation matrix.

Proof
An identity matrix, by definition, has instances of $1$ on the main diagonal and $0$ elsewhere.

Each diagonal element is by definition on one row and one column of the matrix.

Also by definition, each diagonal element is on a different row and column from each other diagonal element.

The result follows by definition of permutation matrix.