grandes-ecoles 2011 QIII.A.2

grandes-ecoles · France · centrale-maths2__mp Matrices Determinant and Rank Computation

For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$ We also denote $\Delta_n = \operatorname{det}(H_n)$.
Show the relation: $$\Delta_{n+1} = \frac{(n!)^4}{(2n)!(2n+1)!} \Delta_n$$
Hint: you may start by subtracting the last column of $\Delta_{n+1}$ from all the others.

For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by:
$$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$
We also denote $\Delta_n = \operatorname{det}(H_n)$.

Show the relation:
$$\Delta_{n+1} = \frac{(n!)^4}{(2n)!(2n+1)!} \Delta_n$$

Hint: you may start by subtracting the last column of $\Delta_{n+1}$ from all the others.

Paper Questions

QI.A.1 QI.A.2 QI.B.1 QI.B.2 QI.B.3 QI.C QI.D QII.A QII.B QII.C QII.D QII.E QII.F QIII.A.1 QIII.A.2 QIII.A.3 QIII.A.4 QIII.A.5 QIII.B.1 QIII.B.2 QIII.B.3 QIII.B.4 QIII.B.5 QIV.A.1 QIV.A.2 QIV.A.3 QIV.A.4 QIV.A.5 QIV.A.6 QIV.B.1 QIV.B.2 QIV.B.3