I need your help please, we have the system X(t+1)= \begin{bmatrix} -1 & \frac{2+(-1)^{t}}{2} \\ \frac{2+(-1)^{t}}{2} & -1 \end{bmatrix} X(t)
By using this formula $\phi(t)=A(t-1)...A(1)A(0)$ , with A(t) is the matrix given in the system
I must prove that $\phi(t)=\frac{1}{2^{t+1}}\begin{bmatrix}(\sqrt{3})^{t}+ (-\sqrt{3})^{t} & (\sqrt{3})^{t+1}+ (-\sqrt{3})^{t+1} \\ (\sqrt{3})^{t+1}+ (-\sqrt{3})^{t+1} & (\sqrt{3})^{t}+ (-\sqrt{3})^{t} \end{bmatrix}$
Unless there's a typo somewhere, the last identity is not true. If $X(t +1) = A(t) X(t)$, then
$$ A(0) = \begin{pmatrix} \color{red}{-1} & 1 \\ 1 & -1 \end{pmatrix}, $$
and
$$ \phi(1) = A(0) = \begin{pmatrix} \color{red}{0} & \bullet \\ \bullet & \bullet \end{pmatrix} $$
even without evaluating the remaining components you see that there's a problem