Eigenvectors and eigenvalues

Let $A$ be an $n\times n$-matrix, and assume $\lambda$ is an eigenvalue of $A$. Show that $\lambda^{2}$ is an eigenvalue of $A^{2}$.

Let $A$ be an invertible $n\times n$-matrix, and assume $\lambda$ is an eigenvalue of $A$. Show that $\lambda\neq 0$ and that $\lambda^{-1}$ is an eigenvalue of $A^{-1}$.

If $A$ is an $n\times n$-matrix and $c$ is a non-zero constant, compare the eigenvalues of $A$ and $cA$.

Let $A,B$ be invertible $n\times n$-matrices which commute. That is, $AB=BA$. Suppose $\mathbf{v}$ is an eigenvector of $B$. Show that then $A\mathbf{v}$ must also be an eigenvector for $B$.

Suppose $A$ is an $n\times n$-matrix and it satisfies $A^{m}=A$ for some $m$ a positive integer larger than 1. Show that if $\lambda$ is an eigenvalue of $A$ then $\lambda$ equals either $0$, $1$, or $-1$.

Show that if $A\mathbf{v}=\lambda\mathbf{v}$ and $A\mathbf{w}=\lambda\mathbf{w}$, then whenever $k,p$ are scalars,

Does this imply that $k\mathbf{v}+p\mathbf{w}$ is an eigenvector? Explain.