grandes-ecoles 2011 QIII.A.1

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 $\Delta_n = \operatorname{det}(H_n)$.
Calculate $H_2$ and $H_3$. Show that these are invertible matrices and determine their inverses.

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

Calculate $H_2$ and $H_3$. Show that these are invertible matrices and determine their inverses.

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