grandes-ecoles 2011 QIII.A.3

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 $\Delta_n = \operatorname{det}(H_n)$.
Using the relation $\Delta_{n+1} = \frac{(n!)^4}{(2n)!(2n+1)!} \Delta_n$, deduce the expression of $\Delta_n$ as a function of $n$ (we will use the quantities $c_m = \prod_{i=1}^{m-1} i!$ for appropriate integers $m$).

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 $\Delta_n = \operatorname{det}(H_n)$.

Using the relation $\Delta_{n+1} = \frac{(n!)^4}{(2n)!(2n+1)!} \Delta_n$, deduce the expression of $\Delta_n$ as a function of $n$ (we will use the quantities $c_m = \prod_{i=1}^{m-1} i!$ for appropriate integers $m$).

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