91Ó°ÊÓ

Let \(A\) be an \(n \times n\) nonzero matrix satisfying \(A^{10}=O .\) Explain why \(A\) must be singular. What properties of determinants are you using in your argument?

If \(A\) is an idempotent matrix \(\left(A^{2}=A\right),\) then prove that the determinant of \(A\) is either 0 or 1