grandes-ecoles

Papers (191)
2025
centrale-maths1__official 40 centrale-maths2__official 42 mines-ponts-maths1__mp 20 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 24 mines-ponts-maths2__psi 26 polytechnique-maths-a__mp 27 polytechnique-maths__fui 16 polytechnique-maths__pc 27 x-ens-maths-a__mp 18 x-ens-maths-c__mp 9 x-ens-maths-d__mp 38 x-ens-maths__pc 27 x-ens-maths__psi 38
2024
centrale-maths1__official 28 centrale-maths2__official 29 geipi-polytech__maths 9 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 19 mines-ponts-maths2__mp 23 mines-ponts-maths2__pc 21 mines-ponts-maths2__psi 21 polytechnique-maths-a__mp 44 polytechnique-maths-b__mp 37 x-ens-maths-a__mp 43 x-ens-maths-b__mp 35 x-ens-maths-c__mp 22 x-ens-maths-d__mp 45 x-ens-maths__pc 24 x-ens-maths__psi 26
2023
centrale-maths1__official 44 centrale-maths2__official 33 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 15 mines-ponts-maths1__pc 23 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 18 mines-ponts-maths2__psi 22 polytechnique-maths__fui 23 x-ens-maths-a__mp 25 x-ens-maths-b__mp 24 x-ens-maths-c__mp 20 x-ens-maths-d__mp 20 x-ens-maths__pc 18 x-ens-maths__psi 15
2022
centrale-maths1__mp 48 centrale-maths1__official 48 centrale-maths1__pc 37 centrale-maths1__psi 43 centrale-maths2__mp 32 centrale-maths2__official 32 centrale-maths2__pc 39 centrale-maths2__psi 45 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 24 mines-ponts-maths1__psi 24 mines-ponts-maths2__mp 24 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 x-ens-maths-a__mp 13 x-ens-maths-b__mp 40 x-ens-maths-c__mp 27 x-ens-maths-d__mp 46 x-ens-maths1__mp 13 x-ens-maths2__mp 40 x-ens-maths__pc 15 x-ens-maths__pc_cpge 15 x-ens-maths__psi 22 x-ens-maths__psi_cpge 23
2021
centrale-maths1__mp 40 centrale-maths1__official 40 centrale-maths1__pc 36 centrale-maths1__psi 29 centrale-maths2__mp 30 centrale-maths2__official 29 centrale-maths2__pc 38 centrale-maths2__psi 37 x-ens-maths2__mp 39 x-ens-maths__pc 44
2020
centrale-maths1__mp 42 centrale-maths1__official 42 centrale-maths1__pc 36 centrale-maths1__psi 40 centrale-maths2__mp 38 centrale-maths2__official 38 centrale-maths2__pc 40 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 24 mines-ponts-maths2__mp_cpge 21 x-ens-maths-a__mp_cpge 18 x-ens-maths-b__mp_cpge 20 x-ens-maths-d__mp 14 x-ens-maths1__mp 18 x-ens-maths2__mp 20 x-ens-maths__pc 18
2019
centrale-maths1__mp 37 centrale-maths1__official 37 centrale-maths1__pc 40 centrale-maths1__psi 39 centrale-maths2__mp 37 centrale-maths2__official 37 centrale-maths2__pc 39 centrale-maths2__psi 49 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 26
2018
centrale-maths1__mp 47 centrale-maths1__official 47 centrale-maths1__pc 41 centrale-maths1__psi 44 centrale-maths2__mp 44 centrale-maths2__official 44 centrale-maths2__pc 35 centrale-maths2__psi 38 x-ens-maths1__mp 19 x-ens-maths2__mp 17 x-ens-maths__pc 22 x-ens-maths__psi 24
2017
centrale-maths1__mp 45 centrale-maths1__official 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__official 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 26 x-ens-maths2__mp 16 x-ens-maths__pc 18 x-ens-maths__psi 26
2016
centrale-maths1__mp 42 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 47 centrale-maths2__psi 27 x-ens-maths1__mp 18 x-ens-maths2__mp 46 x-ens-maths__pc 15 x-ens-maths__psi 20
2015
centrale-maths1__mp 42 centrale-maths1__pc 18 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 18 centrale-maths2__psi 33 x-ens-maths1__mp 16 x-ens-maths2__mp 31 x-ens-maths__pc 30 x-ens-maths__psi 22
2014
centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 27 centrale-maths2__mp 24 centrale-maths2__pc 26 centrale-maths2__psi 27 x-ens-maths1__mp 9 x-ens-maths2__mp 16 x-ens-maths__pc 4 x-ens-maths__psi 24
2013
centrale-maths1__mp 22 centrale-maths1__pc 45 centrale-maths1__psi 29 centrale-maths2__mp 31 centrale-maths2__pc 52 centrale-maths2__psi 32 x-ens-maths1__mp 24 x-ens-maths2__mp 35 x-ens-maths__pc 22 x-ens-maths__psi 9
2012
centrale-maths1__mp 36 centrale-maths1__pc 28 centrale-maths1__psi 33 centrale-maths2__mp 27 centrale-maths2__psi 18
2011
centrale-maths1__mp 27 centrale-maths1__pc 17 centrale-maths1__psi 24 centrale-maths2__mp 29 centrale-maths2__pc 17 centrale-maths2__psi 10
2010
centrale-maths1__mp 19 centrale-maths1__pc 30 centrale-maths1__psi 13 centrale-maths2__mp 32 centrale-maths2__pc 37 centrale-maths2__psi 27
2020 centrale-maths1__pc

36 maths questions

Q1 Matrices Bilinear and Symplectic Form Properties View
In this question only, $n$ is any non-zero natural integer. Determine $J_{n}^{2}$ and show that $J_{n} \in \mathrm{Sp}_{2n}(\mathbb{R}) \cap \mathcal{A}_{2n}(\mathbb{R})$.
Recall: $J_{n} = \left(\begin{array}{cc} 0_{n,n} & I_{n} \\ -I_{n} & 0_{n,n} \end{array}\right)$, and a matrix $M \in \mathcal{M}_{2n}(\mathbb{R})$ is symplectic if and only if $M^{\top} J_{n} M = J_{n}$.
Q2 Matrices Determinant and Rank Computation View
In the case $n=1$: Show that a matrix of size $2 \times 2$ is symplectic if and only if its determinant equals 1.
Recall: $J_{1} = \left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right)$, and a matrix $M \in \mathcal{M}_{2}(\mathbb{R})$ is symplectic if and only if $M^{\top} J_{1} M = J_{1}$.
Q3 Matrices Bilinear and Symplectic Form Properties View
In the case $n=1$: Let $M$ be an orthogonal matrix of size $2 \times 2$. We denote by $M_{1} = \binom{x_{1}}{x_{2}}$ and $M_{2} = \binom{y_{1}}{y_{2}}$ the two columns of $M$. Show the equivalence $$M \text{ is symplectic } \Longleftrightarrow M_{2} = -J_{1} M_{1}.$$
Q4 Matrices Bilinear and Symplectic Form Properties View
In the case $n=1$: Let $X_{1} \in \mathcal{M}_{2,1}(\mathbb{R})$ have norm 1. Show that the square matrix consisting of columns $X_{1}$ and $-J_{1} X_{1}$ is both orthogonal and symplectic.
Q5 Matrices Diagonalizability and Similarity View
In the case $n=1$: Let $M$ be a matrix of size $2 \times 2$ that is symmetric and symplectic. Show that $M$ is diagonalizable and that its eigenvalues are inverses of each other. Show that there exists a matrix $P$ that is both orthogonal and symplectic such that $P^{-1} M P$ is diagonal.
Q6 Matrices Bilinear and Symplectic Form Properties View
In the case $n=1$: Determine the matrices of size $2 \times 2$ that are both antisymmetric and symplectic and show that they are not diagonalizable in $\mathbb{R}$.
Q7 Matrices Bilinear and Symplectic Form Properties View
Let $K$ be an antisymmetric matrix and $\varphi$ the application from $\left(\mathcal{M}_{2n,1}(\mathbb{R})\right)^{2}$ to $\mathbb{R}$ such that $$\forall (X,Y) \in \left(\mathcal{M}_{2n,1}(\mathbb{R})\right)^{2}, \quad \varphi(X,Y) = X^{\top} K Y.$$ Show that $\varphi$ is a bilinear form on $\mathcal{M}_{2n,1}(\mathbb{R})$.
Q8 Matrices Bilinear and Symplectic Form Properties View
Let $K$ be an antisymmetric matrix and $\varphi$ the application from $\left(\mathcal{M}_{2n,1}(\mathbb{R})\right)^{2}$ to $\mathbb{R}$ such that $$\forall (X,Y) \in \left(\mathcal{M}_{2n,1}(\mathbb{R})\right)^{2}, \quad \varphi(X,Y) = X^{\top} K Y.$$ By computing $\varphi(X,X)^{\top}$ in two ways, show that $\varphi$ is alternating. Show similarly that $\varphi$ is antisymmetric.
Q9 Matrices Bilinear and Symplectic Form Properties View
Throughout the rest of the problem, $K = J_{n}$. For all $X = \left(\begin{array}{c} x_{1} \\ x_{2} \\ \vdots \\ x_{2n} \end{array}\right) \in \mathcal{M}_{2n,1}(\mathbb{R})$ and for all $Y = \left(\begin{array}{c} y_{1} \\ y_{2} \\ \vdots \\ y_{2n} \end{array}\right) \in \mathcal{M}_{2n,1}(\mathbb{R})$, where $\varphi(X,Y) = X^{\top} J_{n} Y$, show the equality $$\varphi(X,Y) = \sum_{k=1}^{n} \left(x_{k} y_{k+n} - x_{k+n} y_{k}\right).$$
Q10 Matrices Bilinear and Symplectic Form Properties View
With $K = J_{n}$ and $\varphi(X,Y) = X^{\top} J_{n} Y$: Show that for all $(i,j) \in \{1,\ldots,2n\}^{2}$, $$\varphi(e_{i}, e_{j}) = \delta_{i+n,j} - \delta_{i,j+n}$$ (one may start with the case where $(i,j) \in \{1,\ldots,n\}^{2}$ then generalize).
Q11 Matrices Bilinear and Symplectic Form Properties View
With $K = J_{n}$ and $\varphi(X,Y) = X^{\top} J_{n} Y$: Show that for all $X \in \mathcal{M}_{2n,1}(\mathbb{R})$, $J_{n} X \in X^{\perp}$ and compute $\varphi(J_{n} X, X)$.
Q12 Matrices Bilinear and Symplectic Form Properties View
With $K = J_{n}$ and $\varphi(X,Y) = X^{\top} J_{n} Y$: If $Y \in \mathcal{M}_{2n,1}(\mathbb{R})$, we denote by $Y^{J_{n}}$ the set of vectors $Z$ of $\mathcal{M}_{2n,1}(\mathbb{R})$ such that $\varphi(Y,Z) = 0$. Show that $X^{J_{n}} = (J_{n} X)^{\perp}$.
Q13 Matrices Bilinear and Symplectic Form Properties View
With $K = J_{n}$ and $\varphi(X,Y) = X^{\top} J_{n} Y$: Let $P$ be a symplectic and orthogonal matrix whose columns are denoted $X_{1}, \ldots, X_{2n}$. Show that, for all $(i,j) \in \{1,\ldots,2n\}^{2}$, $$\left\{\begin{array}{l} \|X_{i}\| = 1 \\ i \neq j \Longrightarrow X_{i} \perp X_{j} \\ \varphi(X_{i}, X_{j}) = \delta_{i+n,j} - \delta_{i,j+n} \end{array}\right.$$
Q14 Matrices Bilinear and Symplectic Form Properties View
With $K = J_{n}$, $\varphi(X,Y) = X^{\top} J_{n} Y$, and $P$ a symplectic and orthogonal matrix with columns $X_{1}, \ldots, X_{2n}$ satisfying the properties of Q13: Show that, for all $i \in \{1,\ldots,n\}$, $X_{i}^{J_{n}} = X_{i+n}^{\perp}$.
Q15 Matrices Bilinear and Symplectic Form Properties View
With $K = J_{n}$, $\varphi(X,Y) = X^{\top} J_{n} Y$, and $P$ a symplectic and orthogonal matrix with columns $X_{1}, \ldots, X_{2n}$ satisfying the properties of Q13: Show that, for all $i \in \{1,\ldots,n\}$, $X_{i+n} = -J_{n} X_{i}$.
Q16 Matrices Determinant and Rank Computation View
Show that the determinant of a symplectic matrix equals either 1 or $-1$.
Recall: A matrix $M \in \mathcal{M}_{2n}(\mathbb{R})$ is symplectic if and only if $M^{\top} J_{n} M = J_{n}$.
Q17 Matrices Matrix Group and Subgroup Structure View
Show that the inverse of a symplectic matrix is a symplectic matrix.
Q18 Matrices Matrix Group and Subgroup Structure View
Show that the product of two symplectic matrices is a symplectic matrix. Is the set $\mathrm{Sp}_{2n}(\mathbb{R})$ a vector subspace of $\mathcal{M}_{2n}(\mathbb{R})$?
Q19 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Let $M \in \mathcal{S}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$. Show that if $\lambda$ is an eigenvalue of $M$, then $1/\lambda$ is also an eigenvalue of $M$. Give an eigenvector associated with it.
Q20 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Let $M \in \mathcal{S}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$. Let $\lambda \in \mathrm{sp}_{\mathbb{R}}(M)$ and $p = \dim E_{\lambda}$. Let $(X_{1}, \ldots, X_{p})$ be a basis of $E_{\lambda}$. Show that $(J_{n} X_{1}, \ldots, J_{n} X_{p})$ is a basis of $E_{1/\lambda}$ and that $$\dim(E_{\lambda}) = \dim(E_{1/\lambda}).$$
Q21 Matrices Projection and Orthogonality View
Let $Y_{1}, \ldots, Y_{p}$ be vectors of $\mathcal{M}_{2n,1}(\mathbb{R})$. Let $Y \in \mathcal{M}_{2n,1}(\mathbb{R})$. Show the implication $$Y \in \left(\operatorname{Vect}(Y_{1}, \ldots, Y_{p}, J_{n} Y_{1}, \ldots, J_{n} Y_{p})\right)^{\perp} \Longrightarrow J_{n} Y \in \left(\operatorname{Vect}(Y_{1}, \ldots, Y_{p}, Y, J_{n} Y_{1}, \ldots, J_{n} Y_{p})\right)^{\perp}.$$
Q22 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Let $M \in \mathcal{S}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$. In this question $\lambda = 1$. Show that $E_{1}$ has even dimension and that there exists a basis of $E_{1}$ that is orthonormal and of the form $(X_{1}, \ldots, X_{p}, J_{n} X_{1}, \ldots, J_{n} X_{p})$ where $2p$ is the dimension of $E_{1}$.
Q23 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Let $M \in \mathcal{S}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$. What about $E_{-1}$? (i.e., does $E_{-1}$ have even dimension and does there exist an orthonormal basis of $E_{-1}$ of the form $(X_{1}, \ldots, X_{p}, J_{n} X_{1}, \ldots, J_{n} X_{p})$?)
Q24 Matrices Diagonalizability and Similarity View
Prove the following property: if $M \in \mathcal{S}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$, there exists $P \in \mathcal{O}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$ such that $P^{\top} M P$ is diagonal with diagonal coefficients $d_{1}, \ldots, d_{2n}$ satisfying for all $k \in \{1, \ldots, n\}$, $d_{k+n} = 1/d_{k}$.
Q25 Matrices Bilinear and Symplectic Form Properties View
Let $$A = \frac{1}{8} \left(\begin{array}{llll} 9 & 1 & 3 & 3 \\ 1 & 9 & 3 & 3 \\ 3 & 3 & 9 & 1 \\ 3 & 3 & 1 & 9 \end{array}\right).$$ Show that $A \in \mathcal{S}_{4}(\mathbb{R}) \cap \mathrm{Sp}_{4}(\mathbb{R})$.
Q26 Matrices Matrix Decomposition and Factorization View
Let $$A = \frac{1}{8} \left(\begin{array}{llll} 9 & 1 & 3 & 3 \\ 1 & 9 & 3 & 3 \\ 3 & 3 & 9 & 1 \\ 3 & 3 & 1 & 9 \end{array}\right).$$ Construct an orthogonal and symplectic matrix $P$ such that $P^{\top} A P$ is diagonal.
Q27 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Let $M \in \mathcal{A}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$. Let $m$ be the linear map canonically associated with $M$. Show the equality $\mathrm{sp}_{\mathbb{R}}(M) = \emptyset$.
Q28 Matrices Diagonalizability and Similarity View
Let $M \in \mathcal{A}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$. Show that there exists $P \in \mathcal{O}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$ such that $P^{\top} M^{2} P$ is diagonal with diagonal coefficients $d_{1}, \ldots, d_{2n}$ satisfying for all $k \in \{1, \ldots, n\}$, $d_{k+n} = 1/d_{k}$.
Q29 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Let $M \in \mathcal{A}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$, and let $m$ be the linear map canonically associated with $M$. Let $X$ denote an eigenvector of $M^{2}$ of norm 1 associated with a certain eigenvalue $\lambda$. Show that $MX$, $J_{n} X$ and $J_{n} MX$ are eigenvectors of $M^{2}$ and give the eigenvalues associated with each of these vectors.
Q30 Matrices Linear Transformation and Endomorphism Properties View
Let $M \in \mathcal{A}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$, and let $m$ be the linear map canonically associated with $M$. Let $X$ denote an eigenvector of $M^{2}$ of norm 1 associated with a certain eigenvalue $\lambda$. Denote $F = \operatorname{Vect}(X, MX, J_{n} X, J_{n} MX)$. Show that $F$ is stable under $M$ and under $J_{n}$.
Q31 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Let $M \in \mathcal{A}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$. Show that all eigenvalues of $M^{2}$ are strictly negative.
Q32 Matrices Matrix Decomposition and Factorization View
Let $M \in \mathcal{A}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$, and let $m$ be the linear map canonically associated with $M$. Let $X$ denote an eigenvector of $M^{2}$ of norm 1 associated with a certain eigenvalue $\lambda$, and let $F = \operatorname{Vect}(X, MX, J_{n} X, J_{n} MX)$. Justify that if $\lambda \neq -1$, $F$ is a vector space of dimension 4. Show that, in this case, $$\left(X,\ \frac{-1}{\sqrt{-\lambda}} MX,\ -J_{n} X,\ \frac{1}{\sqrt{-\lambda}} J_{n} MX\right)$$ is an orthonormal basis of $F$. Then give the matrix of the application $m_{F}$ induced by $m$ on $F$ in the basis obtained.
Q33 Matrices Projection and Orthogonality View
Let $M \in \mathcal{A}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$, and let $m$ be the linear map canonically associated with $M$. Let $X$ denote an eigenvector of $M^{2}$ of norm 1 associated with a certain eigenvalue $\lambda$, and let $F = \operatorname{Vect}(X, MX, J_{n} X, J_{n} MX)$. Show that $F^{\perp}$ is stable under $M$ and under $J_{n}$.
Q34 Matrices Linear Transformation and Endomorphism Properties View
Let $M \in \mathcal{A}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$, and let $m$ be the linear map canonically associated with $M$. Show that there exists a non-zero natural integer $q$ and vector subspaces of $\mathcal{M}_{2n,1}(\mathbb{R})$, denoted $F_{1}, \ldots, F_{q}$ such that
  • [(a)] $F_{1} \oplus \cdots \oplus F_{q} = \mathcal{M}_{2n,1}(\mathbb{R})$;
  • [(b)] $\forall i \in \{1,\ldots,q\}$, $F_{i}$ is stable under $M$ and under $J_{n}$;
  • [(c)] $\forall i \in \{1,\ldots,q\}$, $F_{i}^{\perp}$ is stable under $M$ and under $J_{n}$;
  • [(d)] $\forall (i,j) \in \{1,\ldots,q\}^{2}$, $i \neq j \Longrightarrow \forall (Y,Z) \in F_{i} \times F_{j}$, $\langle Y,Z \rangle = 0 = \varphi(Y,Z)$;
  • [(e)] $\forall i \in \{1,\ldots,q\}$, $\dim F_{i} \in \{2,4\}$;
  • [(f)] $\forall i \in \{1,\ldots,q\}$, the matrix of the application $m_{F_{i}}$ induced by $m$ on $F_{i}$ in a certain basis is of the form $$J_{1} \quad \text{or} \quad \left(\begin{array}{cc} \sqrt{-\lambda} J_{1} & 0_{2,2} \\ 0_{2,2} & \frac{1}{\sqrt{-\lambda}} J_{1} \end{array}\right).$$
Q35 Matrices Matrix Power Computation and Application View
Let $$B = \frac{1}{4} \left(\begin{array}{cccc} 0 & -5 & 0 & -3 \\ 5 & 0 & 3 & 0 \\ 0 & -3 & 0 & -5 \\ 3 & 0 & 5 & 0 \end{array}\right).$$ Compute $B^{2} \left(\begin{array}{l} 1 \\ 1 \\ 1 \\ 1 \end{array}\right)$.
Q36 Matrices Matrix Decomposition and Factorization View
Let $$B = \frac{1}{4} \left(\begin{array}{cccc} 0 & -5 & 0 & -3 \\ 5 & 0 & 3 & 0 \\ 0 & -3 & 0 & -5 \\ 3 & 0 & 5 & 0 \end{array}\right).$$ Determine a real number $a$ and a matrix $P$ such that $$P \in \mathcal{O}_{4}(\mathbb{R}) \cap \mathrm{Sp}_{4}(\mathbb{R}) \quad \text{and} \quad P^{\top} B P = \left(\begin{array}{cccc} 0 & a & 0 & 0 \\ -a & 0 & 0 & 0 \\ 0 & 0 & 0 & 1/a \\ 0 & 0 & -1/a & 0 \end{array}\right).$$