In this subsection, we assume that $J_n = J_n^{(1)}$, the matrix introduced in subsection A-IV. We adopt the following convention: for all $x = (x_1, \ldots, x_n) \in \Lambda_n$, we denote $x_{n+1} = x_1$ and $x_0 = x_n$. We set $A = \begin{pmatrix} \mathrm{e}^{\beta - h} & \mathrm{e}^{-\beta - h} \\ \mathrm{e}^{-\beta + h} & \mathrm{e}^{\beta + h} \end{pmatrix}$. Show that $Z_n(h) = \operatorname{tr}(A^n)$, where $\operatorname{tr}$ denotes the trace of a square matrix.
We assume in this part that all roots of $p$ are stable and have multiplicity 1 and we denote by $h = Xp'$ (where $p'$ is the derivative polynomial of $p$) and $h_0$ the reciprocal polynomial of $h$. We recall that, according to question 3, there exists a real number $\lambda \in \{-1, 1\}$ such that $p = \lambda p_0$. Show that there exists a real number $\eta > 0$ such that for every $r \in ]1-\eta; 1[$, the polynomial $p(rX)$ is split, has exactly $\sigma(p)$ roots inside the interval $]-1; 1[$ and has no stable root.
In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II. For all $x \in \Lambda_n$, we set $s_n(x) = \sum_{k=1}^n x_i$. Verify that $$Z_n(h) = \mathrm{e}^{-\frac{\beta}{2}} \sum_{x \in \Lambda_n} \exp\left(\frac{\beta}{2n}\left(s_n(x)\right)^2 + h s_n(x)\right).$$
Let $n \geq 1$ be an integer. We say that a matrix $M \in \mathcal{M}_n(\mathbb{R})$ is doubly stochastic if for all $i, j \in \{1, \ldots, n\}$ we have $$M_{ij} \geq 0 \quad \text{and} \quad \sum_{k=1}^n M_{ik} = \sum_{k=1}^n M_{kj} = 1.$$ We denote by $B_n$ the set of doubly stochastic matrices in $\mathcal{M}_n(\mathbb{R})$. Show that $B_n$ is a polytope and determine its dimension.
Let $n \geq 1$ be an integer. We denote by $B_n$ the set of doubly stochastic matrices in $\mathcal{M}_n(\mathbb{R})$ and $S_n$ the symmetric group of order $n$. For all $\sigma \in S_n$, we define $P^\sigma \in \mathcal{M}_n(\mathbb{R})$ as follows: for $i, j \in \{1, 2, \ldots, n\}$ we set $P^\sigma_{ij} = 1$ if $j = \sigma(i)$, $P^\sigma_{ij} = 0$ otherwise. Show that $P^\sigma$ is a vertex of $B_n$ for all $\sigma \in S_n$.
Let $n \geq 1$ be an integer. We denote by $B_n$ the set of doubly stochastic matrices in $\mathcal{M}_n(\mathbb{R})$ and $\mathcal{M}_n(\mathbb{Z})$ the set of $n \times n$ matrices with integer coefficients. Suppose that $M \in B_n \backslash \mathcal{M}_n(\mathbb{Z})$. Show that there exists a sequence $(r_1, s_1), (r_2, s_2), \ldots, (r_k, s_k)$ of pairs of indices with $k \geq 2$ such that $$0 < M_{r_i, s_i} < 1, \quad 0 < M_{r_i, s_{i+1}} < 1 \quad \text{and} \quad (r_k, s_k) = (r_1, s_1)$$ then that we can assume that all the pairs $(r_1, s_1), (r_1, s_2), (r_2, s_2), \ldots, (r_{k-1}, s_{k-1}), (r_{k-1}, s_k)$ are distinct.
140- If $A = \begin{bmatrix} 0 & -\tan\alpha \\ \tan\alpha & 0 \end{bmatrix}$ and $I$ is the identity matrix of order 2, the first row of $(I+A)(I-A)^{-1}$ is which of the following? (1) $[\cos 2\alpha \;\; -\sin 2\alpha]$ (2) $[\cos 2\alpha \;\; \sin 2\alpha]$ (3) $[\sin 2\alpha \;\; \cos 2\alpha]$ (4) $[-\sin 2\alpha \;\; \cos 2\alpha]$
139- If matrix $A$ has the transformation $T(x,y) = (2x - y,\ 3x - 4y)$ and $I$ is the identity matrix, and $\alpha$ and $\beta$ are two real numbers such that $\alpha A + \beta I = A^{-1}$, what is the value of $\beta$? (1) $-\dfrac{3}{5}$ (2) $-\dfrac{1}{5}$ (3) $\dfrac{2}{5}$ (4) $\dfrac{4}{5}$
139- If $A = [a_{ij}]_{r \times 3}$ and $B = [b_{ij}]_{r \times 2}$, which of the following matrix products is defined? (1) $AB$ (2) $A^t B$ (3) $B^t A^t$ (4) $AB^t$
140- If $A = \begin{bmatrix} 1 & -1 & 2 \\ 0 & 2 & -1 \\ 0 & 0 & 3 \end{bmatrix}$, what is the sum of the entries of the second column of $A^{-1}$? (1) $-\dfrac{1}{3}$ (2) $\dfrac{2}{3}$ (3) $1$ (4) zero
137. Matrix $A = \begin{bmatrix} 5 & 2 & -1 \\ 4 & 3 & -2 \\ 1 & 6 & 7 \end{bmatrix}$ is written as the sum of a symmetric matrix and a skew-symmetric matrix. The determinant of the symmetric matrix is: (1) $16$ (2) $18$ (3) $22$ (4) $24$
140-- Three pages with matrix equations. If $\begin{vmatrix} a & -1 & 3 \\ b & 2 & 4 \\ c & -2 & 1 \end{vmatrix} = 5$, then the three pages intersect. With which of the following lengths do they intersect? $$\begin{bmatrix} 2 & -1 & 3 \\ 1 & 2 & 4 \\ 3 & -2 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} a \\ b \\ c \end{bmatrix}$$ (1) $-\dfrac{1}{3}$ (2) $-\dfrac{1}{2}$ (3) $\dfrac{1}{3}$ (4) $\dfrac{1}{2}$
132- If $A$ is a $3 \times 3$ matrix and $|A| = 4$, then the determinant of matrix $A \cdot A$ is which of the following? (1) $64$ (2) $96$ (3) $128$ (4) $256$
138- If $A = \begin{bmatrix} x & -1 & -x \\ 0 & 0 & 4 \\ y & z & z \end{bmatrix}$, $B = \begin{bmatrix} yz & \frac{1}{2} & 2 \\ yz & 0 & -4y \\ 0 & \frac{1}{2} & 0 \end{bmatrix}$ and matrix $AB$ is scalar for every $y \in \mathbb{Z}$, the value of $xy$ is which? \[
\text{(1)}\ -1 \qquad \text{(2)}\ -2 \qquad \text{(3)}\ 1 \qquad \text{(4)}\ 2
\]
A ``basic row operation'' on a matrix means adding a multiple of one row to another row. Consider the matrices $$A = \left(\begin{array}{rrr} x & 5 & x \\ 1 & 3 & -2 \\ -2 & -2 & 2 \end{array}\right) \quad \text{and} \quad B = \left(\begin{array}{rrr} 0 & 0 & 21 \\ 1 & -1 & -14 \\ 0 & \frac{4}{3} & 4 \end{array}\right)$$ It is given that $B$ can be obtained from $A$ by applying finitely many basic row operations. Then, the value of $x$ is: (A) 3 (B) $-3$ (C) $-1$ (D) 2.