= Solution
For either eigenpair above, $\lambda_i\in\{0,1\}$ and hence $\lambda_i^n=\lambda_i$ for every positive integer $n$. Therefore
$$
\frac{d^n}{dt^n}(u_i e^{\lambda_it})
=\lambda_i^n u_i e^{\lambda_it}
=A u_i e^{\lambda_it},
$$
so both terms in the solution from part (a), and every linear combination of them, solve the higher-order system.
A system of two scalar differential equations of order $n$ has a $2n$-dimensional solution space. The displayed family supplies two independent solutions, so there must be
$$
\boxed{2n-2}
$$
further linearly independent solutions.
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