Chapter 4.5, Problem 1E

The following list of matrices and their respective characteristic polynomials is referred to in Exercises 1-11.

$A=\left[\begin{array}{cc}2& -1\\ -1& 2\end{array}\right]$, $p\left(t\right)=\left(t-3\right)(t-1)$,	$B=\left[\begin{array}{cc}1& -1\\ 1& 3\end{array}\right]$, $p\left(t\right)={\left(t-2\right)}^2$,
$C=\left[\begin{array}{ccc}-6& -1& 2\\ 3& 2& 0\\ -14& -2& 5\end{array}\right]$, $p\left(t\right)=-{\left(t-1\right)}^2(t+1)$,	$D=\left[\begin{array}{ccc}-7& 4& -3\\ 8& -3& 3\\ 32& -16& 13\end{array}\right]$, $p\left(t\right)=-{\left(t-1\right)}^3$,
$E=\left[\begin{array}{cccc}6& 4& 4& 1\\ 4& 6& 1& 4\\ 4& 1& 6& 4\\ 1& 4& 4& 6\end{array}\right]$, $p\left(t\right)=\left(t+1\right){\left(t+5\right)}^2\left(t-15\right)$,	$F=\left[\begin{array}{cccc}1& -1& -1& -1\\ -1& 1& -1& -1\\ -1& -1& 1& -1\\ -1& -1& -1& 1\end{array}\right]$, $p\left(t\right)=\left(t+2\right){\left(t-2\right)}^3$

In Exercises 1-11, find a basis for the eigenspace $E_{\lambda }$ for the given matrix and the value of $\lambda$. Determine the algebraic and geometric multiplicities of $\lambda$.

$A,\lambda =3$