>>11222066> Fact: all permutation matrices are involutary.Not true. E.g. consider the matrix [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]] which maps [a,b,c,d]^T -> [b,c,d,a]^T. This isn't involutory, although its square is; the matrix itself satisfies A^4=I => (A^2)^2=I. A permutation matrix which swaps 2 rows or columns is involutory, as is one which swaps any number of disjoint pairs, but a permutation with cycles larger than 2 isn't.
> Fact: all involutary matrices A have det(A) =1 or =-1.This follows from det(AB)=det(A)*det(B) and det(I)=1.
A matrix which is its own inverse satisfies A.A=I and as such can be considered a square root of the identity matrix; there are infinitely many of them.
Some more examples of matrices which are their own inverse (i.e. A^2=I) include a reflection in any plane (for which the determinant is always -1), or a 3x3 matrix corresponding to a rotation of 180° about any axis (for which the determinant is always 1).
Similarly, a rotation of 360°/N about any axis can be considered an Nth root of the identity matrix.
