Let $M \in \mathscr{M}_d\left(\mathbb{R}_{+}\right)$. We assume that the matrix $M$ has an eigenvalue $\lambda > 0$ and that there exists $h \in \mathscr{M}_{d,1}\left(\mathbb{R}_{+}^{*}\right)$ a column vector such that: $$Mh = \lambda h.$$ We also assume that there exist $\nu \in \mathscr{P}$ and $c > 0$ such that for all $i,j \in \{1,\ldots,d\}$, $$M_{i,j} \geqslant c\nu_j.$$ We introduce the matrix $P \in \mathscr{M}_d\left(\mathbb{R}_{+}\right)$ defined for $1 \leqslant i,j \leqslant d$ by $$P_{i,j} = \frac{M_{i,j} h_j}{\lambda h_i}.$$
Justify that for all $i \in \{1,\ldots,d\}$, $\displaystyle\sum_{j=1}^{d} P_{i,j} = 1$.

We assume in this question only that $n = 2$. Determine $u(A)$ in the following case: $$A = \begin{pmatrix} \alpha & \gamma \\ 0 & \beta \end{pmatrix}$$ where $\alpha, \beta$ and $\gamma$ are fixed real numbers with $\alpha \neq \beta$ and $\{\alpha, \beta\} \subset D_u$. We will express the coefficients of $u(A)$ in terms of $\alpha, \beta$ and $\gamma, U(\alpha)$ and $U(\beta)$.

Let $m \in \mathbb{N}$. We consider the matrix $$M = \left(\begin{array}{cccccc} \binom{0}{0} & 0 & \cdots & \cdots & \cdots & 0 \\ \binom{1}{0} & \binom{1}{1} & 0 & & & \vdots \\ \vdots & & \ddots & \ddots & & \vdots \\ \vdots & & & \ddots & \ddots & \vdots \\ \binom{m-1}{0} & & & & \binom{m-1}{m-1} & 0 \\ \binom{m}{0} & \cdots & \cdots & \cdots & \cdots & \binom{m}{m} \end{array}\right) \in \mathscr{M}_{m+1}(\mathbb{R}).$$ Deduce that for all $\left(u_{0}, \ldots, u_{m}\right), \left(v_{0}, \ldots, v_{m}\right) \in \mathbb{R}^{m+1}$, $$\text{if} \quad \forall k \leqslant m, \quad u_{k} = \sum_{\ell=0}^{k} \binom{k}{\ell} v_{\ell}, \quad \text{then} \quad \forall k \leqslant m, \quad v_{k} = \sum_{\ell=0}^{k} (-1)^{k-\ell} \binom{k}{\ell} u_{\ell}$$

Let $m \in \mathbb{N}$. We consider the matrix $$M = \left(\begin{array}{cccccc} \binom{0}{0} & 0 & \cdots & \cdots & \cdots & 0 \\ \binom{1}{0} & \binom{1}{1} & 0 & & & \vdots \\ \vdots & & \ddots & \ddots & & \vdots \\ \vdots & & & \ddots & \ddots & \vdots \\ \binom{m-1}{0} & & & & \binom{m-1}{m-1} & 0 \\ \binom{m}{0} & \cdots & \cdots & \cdots & \cdots & \binom{m}{m} \end{array}\right) \in \mathscr{M}_{m+1}(\mathbb{R}).$$ Deduce that for all $\left(u_0, \ldots, u_m\right), \left(v_0, \ldots, v_m\right) \in \mathbb{R}^{m+1}$, $$\text{if} \quad \forall k \leqslant m, \quad u_k = \sum_{\ell=0}^{k} \binom{k}{\ell} v_\ell, \quad \text{then} \quad \forall k \leqslant m, \quad v_k = \sum_{\ell=0}^{k} (-1)^{k-\ell} \binom{k}{\ell} u_\ell.$$

We set $N = n^2$ and $$J_N^{(2)} = I_n \otimes J_n^{(1)} + J_n^{(1)} \otimes I_n \in \mathcal{M}_N(\mathbb{R})$$
Verify that, in the case where $n = 3$, $J_N^{(2)}$ is the matrix such that, for all $(i,j) \in \llbracket 1,9 \rrbracket^2$, the coefficient with index $(i,j)$ equals 1 if the vertices $i$ and $j$ of the graph are connected by an edge (the edges count whether they are dashed or solid), and equals 0 otherwise.

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 $$Z_n(h) = \mathrm{e}^{-\frac{\beta}{2}} \sum_{x \in \Lambda_n} \exp\left(\frac{\beta}{2n}\left(s_n(x)\right)^2 + h s_n(x)\right).$$

138- If $A = \begin{bmatrix} 1 & 3 & 6 & 24 \\ \frac{1}{2} & 1 & 2 & 8 \end{bmatrix}$, $B = \begin{bmatrix} \frac{1}{6} & \frac{1}{2} & 1 & 4 \\ \frac{1}{24} & \frac{1}{8} & \frac{1}{4} & 1 \end{bmatrix}$, and $C = \begin{bmatrix} A \\ B \end{bmatrix}$, the sum of the main diagonal entries of matrix $C^T$ is which of the following?
(1) $16$ (2) $18$ (3) $20$ (4) $24$

Let $\alpha$ and $\beta$ be the distinct roots of the equation $x ^ { 2 } + x - 1 = 0$. Consider the set $T = \{ 1 , \alpha , \beta \}$. For a $3 \times 3$ matrix $M = \left( a _ { i j } \right) _ { 3 \times 3 }$, define $R _ { i } = a _ { i 1 } + a _ { i 2 } + a _ { i 3 }$ and $C _ { j } = a _ { 1 j } + a _ { 2 j } + a _ { 3 j }$ for $i = 1,2,3$ and $j = 1,2,3$.
Match each entry in List-I to the correct entry in List-II.
List-I
(P) The number of matrices $M = \left( a _ { i j } \right) _ { 3 \times 3 }$ with all entries in $T$ such that $R _ { i } = C _ { j } = 0$ for all $i , j$, is
(Q) The number of symmetric matrices $M = \left( a _ { i j } \right) _ { 3 \times 3 }$ with all entries in $T$ such that $C _ { j } = 0$ for all $j$, is
(R) Let $M = \left( a _ { i j } \right) _ { 3 \times 3 }$ be a skew symmetric matrix such that $a _ { i j } \in T$ for $i > j$. Then the number of elements in the set $\left\{ \left( \begin{array} { l } x \\ y \\ z \end{array} \right) : x , y , z \in \mathbb { R } , M \left( \begin{array} { l } x \\ y \\ z \end{array} \right) = \left( \begin{array} { c } a _ { 12 } \\ 0 \\ - a _ { 23 } \end{array} \right) \right\}$ is
(S) Let $M = \left( a _ { i j } \right) _ { 3 \times 3 }$ be a matrix with all entries in $T$ such that $R _ { i } = 0$ for all $i$. Then the absolute value of the determinant of $M$ is
List-II
(1) 1
(2) 12
(3) infinite
(4) 6
(5) 0
The correct option is
(A) $( \mathrm { P } ) \rightarrow ( 4 )$, $( \mathrm { Q } ) \rightarrow ( 2 )$, $( \mathrm { R } ) \rightarrow ( 5 )$, $( \mathrm { S } ) \rightarrow ( 1 )$
(B) $( \mathrm { P } ) \rightarrow ( 2 )$, $( \mathrm { Q } ) \rightarrow ( 4 )$, $( \mathrm { R } ) \rightarrow ( 1 )$, $( \mathrm { S } ) \rightarrow ( 5 )$
(C) $( \mathrm { P } ) \rightarrow ( 2 )$, $( \mathrm { Q } ) \rightarrow ( 4 )$, $( \mathrm { R } ) \rightarrow ( 3 )$, $( \mathrm { S } ) \rightarrow ( 5 )$
(D) $( \mathrm { P } ) \rightarrow ( 1 )$, $( \mathrm { Q } ) \rightarrow ( 5 )$, $( \mathrm { R } ) \rightarrow ( 3 )$, $( \mathrm { S } ) \rightarrow ( 4 )$

Let M denote the set of all real matrices of order $3 \times 3$ and let $\mathrm { S } = \{ - 3 , - 2 , - 1,1,2 \}$. Let
$$\begin{aligned} & \mathrm { S } _ { 1 } = \left\{ \mathrm { A } = \left[ a _ { \mathrm { ij } } \right] \in \mathrm { M } : \mathrm { A } = \mathrm { A } ^ { \mathrm { T } } \text { and } a _ { \mathrm { ij } } \in \mathrm { S } , \forall \mathrm { i } , \mathrm { j } \right\} , \\ & \mathrm { S } _ { 2 } = \left\{ \mathrm { A } = \left[ a _ { \mathrm { ij } } \right] \in \mathrm { M } : \mathrm { A } = - \mathrm { A } ^ { \mathrm { T } } \text { and } a _ { \mathrm { ij } } \in \mathrm { S } , \forall \mathrm { i } , \mathrm { j } \right\} , \\ & \mathrm { S } _ { 3 } = \left\{ \mathrm { A } = \left[ a _ { \mathrm { ij } } \right] \in \mathrm { M } : a _ { 11 } + a _ { 22 } + a _ { 33 } = 0 \text { and } a _ { \mathrm { ij } } \in \mathrm { S } , \forall \mathrm { i } , \mathrm { j } \right\} . \end{aligned}$$
If $n \left( \mathrm { S } _ { 1 } \cup \mathrm { S } _ { 2 } \cup \mathrm { S } _ { 3 } \right) = 125 \alpha$, then $\alpha$ equals

Number of matrices A of order $3 \times 2$ such that all of its elements are from the set $\{ - 2 , - 1,0,1,2 \}$ such that trace of $\mathrm { AA } ^ { \mathrm { T } }$ is 5 , is equal to
(A) 120
(B) 192
(C) 312
(D) 126

Let $A$ be the rotation matrix that rotates counterclockwise by angle $\theta$ about the origin, and let $B$ be the reflection matrix with the $x$-axis as the axis of reflection (axis of symmetry). Let $A = \left[\begin{array}{ll} a_{1} & a_{2} \\ a_{3} & a_{4} \end{array}\right]$ and $BA = \left[\begin{array}{ll} c_{1} & c_{2} \\ c_{3} & c_{4} \end{array}\right]$.
Given that $a_{1} + a_{2} + a_{3} + a_{4} = 2(c_{1} + c_{2} + c_{3} + c_{4})$, then $\tan\theta =$ . (Express as a fraction in lowest terms)