Centrale-Supélec 2026 — Mathématiques 2 (TSI)

Independent solutions, written by Pulsars. Neither official nor affiliated with Concours Centrale-Supélec.

1. Exercise 1 — Partie A · I — Le cas n = 2 (Q1, Q2)

   u(P) = P(X+1) and v(P) = P(X−1) on R_n[X].

   • Q1 — u is an endomorphism of R_2[X]; its matrix in the canonical basis.
   • Q2 — this matrix is invertible; its inverse, its spectrum, its diagonalisability.
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2. Exercise 2 — Partie A · II — Des matrices semblables à leurs inverses (Q3 à Q7)

   U is the matrix of u in the canonical basis of R_n[X].

   • Q3 — U is upper triangular with binomial coefficients; is it diagonalisable?
   • Q4 — u ∘ v, the invertibility of U, and the coefficients of U^{-1}.
   • Q5, Q6 — the matrix D of alternating signs, and the similarity U ∼ U^{-1}.
   • Q7 — the same result for the lower matrix L = U^⊤.
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3. Exercise 3 — Partie A · III — La matrice symétrique de Pascal (Q8 à Q12)

   S = LU.

   • Q8 — the case n = 2: computation of S, diagonalisability, spectrum {1, α, 1/α}.
   • Q9 — the general case: S is diagonalisable, invertible, similar to its inverse.
   • Q10, Q11 — the spectrum of S consists of strictly positive reals, and is stable under x ↦ 1/x.
   • Q12 — is the converse true?
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4. Exercise 4 — Partie A · IV — Les coefficients de S par dénombrement (Q13 à Q15)

   Ω_{i,j} denotes the set of paths from the source s to the vertex (i, j) in the directed graph G_n.

   • Q13 — the cardinality of Ω_{2,2}, then of Ω_{i,j} in general.
   • Q14 — a complete system of events giving Vandermonde's identity.
   • Q15 — the coefficient of index (i, j) of S.
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5. Exercise 5 — Partie B · I — La suite des dérangements (Q16 à Q22)

   a_0 = 1, a_1 = 0, and a_{n+1} = n(a_n + a_{n−1}).

   • Q16, Q17 — the first values, and a Python function.
   • Q18, Q19 — a_n is a natural number, bounded above by n!.
   • Q20 to Q22 — the bounds given by the partial sums of Σ(−1)^k/k!, the identity a_n = n!S_n, and the asymptotic equivalent.
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6. Exercise 6 — Partie B · II — La série entière associée (Q23 à Q25)

   s(x) = Σ (a_n/n!)x^n.

   • Q23 — the radius of convergence equals 1, and s is a solution of a differential system.
   • Q24 — solving it, and the closed form of s.
   • Q25 — Leibniz's formula applied to x ↦ e^x s(x).
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7. Exercise 7 — Partie C · I et II — Représenter une permutation (Q26 à Q30)

   A permutation is represented by a directed graph G(σ) or by a matrix M(σ) whose (i, j) coefficient equals 1 if and only if σ(i) = j.

   • Q26, Q27 — going from one representation to the other on two examples.
   • Q28 to Q30 — the inverse and the square of a permutation matrix.
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8. Exercise 8 — Partie C · III — Matrices de permutation diagonalisables (Q31 à Q35)

   The theorem to be proved: M(σ) is diagonalisable if and only if G(σ) contains only loops or bi-loops.

   • Q31 — the example: characteristic polynomial and diagonalisability.
   • Q32 to Q34 — the forward direction.
   • Q35 — the converse.
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9. Exercise 9 — Partie D — Graphes de permutations sans boucle (Q36 à Q41)

   B_n is the event "the chosen graph has no loop", b_n its cardinality, and X_n the number of loops.

   • Q36, Q37 — the cardinality of G(S_n), the values of X_n, and its distribution.
   • Q38, Q39 — the relation between p! and the b_k, then the first values.
   • Q40, Q41 — the required probability, obtained in two ways.
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