Find a matrix $B$ such that $B^2=A$.. Let $A$ be a $2 × 2$ matrix and $I$ be the identity matrix. Assume that the null spaces of $A − 4I$
and $A − I$ respectively are spanned by
$\begin{bmatrix}3\\2\end{bmatrix}$
and $\begin{bmatrix}1\\1\end{bmatrix}$
respectively. Find a matrix $B$ such that
$B^2 = A$.
How to approach this problem? Any hint. Thanks in advance.

$\begin{bmatrix}3 \\ 2\end{bmatrix}$ and $\begin{bmatrix} 1\\ 1\end{bmatrix}$ are eignevectors of $A$ with eigenvalues 4 and 1, respectively. Now, we just need to make a matrix $B$ such that $\begin{bmatrix} 3\\ 2\end{bmatrix}$ is an eigenvector with eigenvalue $\pm 2$ and $\begin{bmatrix}1\\1\end{bmatrix}$ is an eigenvector with eigenvalue $\pm 1$. I'll leave the rest to you, good luck (:
