Q:

Find $B=A^2+A$ knowing that $A^3=\begin{bmatrix}4&3\\-3&-2\end{bmatrix}$

Find $B=A^2+A$ knowing that $A^3=\begin{bmatrix}4&3\\-3&-2\end{bmatrix}$

Is there a way to solve this rather than just declaring a matrix $$A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$$ and then trying to solve a system of cubic equations? My attempt:
$$A^3 -I_2 = \begin{bmatrix} 3 & 3\\ -3 & -3\end{bmatrix} = 9 \begin{bmatrix} 1 & 1\\ -1 & -1\end{bmatrix}$$
and 
$$B=\frac {A^3-I_2}{A-I_2}-I_2$$
$$(A-I_2)(A^2+A+I_2)=9\begin{bmatrix}1&1\\-1&-1\end{bmatrix}$$
But I get stuck here.

A:

The matrix $C=A^3$ satisfies its characteristic equation, that is: $det(A^3-xI)=0$, which is $(4-x)(-2-x)+9=0$. 
See https://en.wikipedia.org/wiki/Cayley%E2%80%93Hamilton_theorem.
This gives $C^2-2C+I=0$, thus $C$ has the eigenvalue 1. The eigenvalues of $A^3$ are the cubes of the eigenvalues of $A$. If the eigenvalues of $A$ are the two complex cube roots of unity, then we'd get $A^2+A=-I$, but then $A^3-I$ is non-zero, thus the eigenvalues of $A$ are also both 1, which means that $A$ satisfies $A^2=2A-I$. 
This gives $A^3=2A^2+A=2(2A-I)+A=3A-2I$. Thus, $A^2+A=3A-I=A^3+I$.
Remark: The conclusion that the eigenvalues of $A$ must both be 1 is wrong, as they could be any of the other two cube roots of unity, occuring twice. This is shown in the other answers.

A:

Although I prefer an algebraic approach, here is a "non-algebraic" way by directly determining $A$. Let 

$C =A^3 \Rightarrow C^2=\begin{bmatrix}7& 6\\-6&-5\end{bmatrix}$

Looking at the pattern of the entries comparing $A^6$ with $A^3$ we find: 

the entries in the first row go $3$ up
the entries in the second row go $3$ down

So, applying this pattern "backwards" to $A^3$ we get a possible candidate for $A$:
$$A = \begin{bmatrix}2& 1\\-1&0\end{bmatrix}$$ 
Indeed, we get
$$A^3 = \begin{bmatrix}4&3\\-3&-2\end{bmatrix} \Rightarrow A^2+A = \begin{bmatrix}5&3\\-3&-1\end{bmatrix}$$

A:

Presumably $A$ is real. Observe that $A^3-I$ is nonzero but nilpotent (because $(A^3-I)^2=0$). Therefore $A^3$ and in turn $A$ are not diagonalisable. Hence $A$ has repeated eigenvalues. So, if $A$ is real, its eigenvalues must be real (or else the trace of $A$ would become non-real) and equal to $1$ (because $A^3-I$ is nilpotent). Hence the Jordan form of $A$ is $\pmatrix{1&1\\0&1}$. Since
$$
\pmatrix{1&1\\0&1}^2+\pmatrix{1&1\\0&1}=\pmatrix{2&3\\0&2}
=\pmatrix{1&1\\0&1}^3+I,
$$
we conclude that $A^2+A=A^3+I$.
Remark. Note that the above conclusion does not hold when $A$ can be non-real. (Thus the other answers here are either wrong or incomplete.) E.g. suppose $w=\exp(2\pi i/3)$ and
$$
A=\pmatrix{2w&w\\ -w&0}.
$$
Then $A^3$ is indeed equal to $\pmatrix{4&3\\ -3&-2}$ but $(A^2+A)-(A^3+I)$ is non-real.

