### Instruction: Let
\[\mathbf{M} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\]be a matrix with real entries such that $\mathbf{M}^3 = \mathbf{I}.$  Enter all possible values of $a + d,$ separated by commas.### Response: The first thing I notice is that $\mathbf{M}$ is a $2 \times 2$ matrix, and it satisfies $\mathbf{M}^3 = \mathbf{I}.$
This means that $\mathbf{M}$ is invertible, and its inverse is $\mathbf{M}^2.$
I also notice that $\mathbf{M}^3 = \mathbf{I}$ implies that $\mathbf{M}$ has finite order, that is, there is some positive integer $n$ such that $\mathbf{M}^n = \mathbf{I}.$
In fact, since $\mathbf{M}^3 = \mathbf{I},$ it follows that $n$ must be a divisor of $3,$ so either $n = 1$ or $n = 3.$
If $n = 1,$ then $\mathbf{M} = \mathbf{I},$ and $a + d = 2.$
If $n = 3,$ then $\mathbf{M}$ is a rotation matrix, and it rotates the plane by $\frac{2\pi}{3}$ radians.
I can write $\mathbf{M}$ in the form $\begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix},$ where $\theta = \frac{2\pi}{3}.$
Then, $a + d = 2 \cos \theta = 2 \cos \frac{2\pi}{3} = -1.$
Therefore, the possible values of $a + d$ are $-1$ and $2.$