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.

Observe that $A \dbinom{3}{2} = 4 \dbinom{3}{2}$ and that $A \dbinom{1}{1} = \dbinom{1}{1}$. So, if we define $P := \bigg( \begin{matrix} 3&1\\2&1 \end{matrix} \bigg)$ then
$$P^{-1}AP \dbinom{1}{0} = \dbinom{4}{0} \textrm{ and } P^{-1}AP \dbinom{0}{1} = \dbinom{0}{1},$$
that is, $P^{-1}AP = \bigg( \begin{matrix} 4&0\\0&1 \end{matrix} \bigg)$. Now, all that you need is a matrix $B$ such that $(P^{-1}BP)^2 = P^{-1}AP$.
