1. Let M = \begin{pmatrix} 0 & 1 & 0 & \cdots & 0 \\ 1 & \ddots & \ddots & \ddots & \vdots \\ 0 & \ddots & \ddots & \ddots & 0 \\ \vdots & \ddots & \ddots & \ddots & 1 \\ 0 & \cdots & 0 & 1 & 0 \end{pmatrix} \in \mathcal{M}_n(\mathbb{C}).

What are the eigenvalues and eigenspaces of M ?

2. Let n be a prime number. Let ( X_0, \ldots, X_{n-1} ) be independent random variables such that P(X_i = 0) = P(X_i = 1) = \frac{1}{2} .

Let A = \begin{pmatrix} X_0 & X_{n-1} & \cdots & X_1 \\ X_1 & \ddots & \ddots & \vdots \\\vdots & \ddots & \ddots & X_{n-1} \\ X_{n-1} & \cdots & X_1 & X_0\end{pmatrix}

Calculate P(A \notin GL_n(\mathbb{C}))