Question 39572
Let {{{A^4 = A}}}.  Then {{{A^4 -A=O}}}.  (The zero matrix!)Taking the determinants of both sides,
{{{det(A^4 -A)=det(O) = 0}}},
{{{det(A(A^3 - I)) = 0}}},
{{{det(A)*det(A^3 - I) = 0}}},or det(A) = 0, or {{{det(A^3 - I) = 0}}}. 
If A is nonsingular, then det(A)is NOT equal to zero, but {{{A^3 - I= O}}},(because {{{A^3 = A^4*A^-1 = A*A^-1 = I}}}) so {{{det(A^3 - I) = 0}}}. Hence 
{{{A^2 = A^3*A^-1 = I*A^-1 = A^-1}}}.
If A is singular , then det(A) = 0 (since a square matrix is singular if and only if its determinant is 0). In which case the value of  {{{det(A^3 - I)}}} won't matter anymore.