grandes-ecoles 2016 QII.B.4

grandes-ecoles · France · centrale-maths1__psi Matrices Determinant and Rank Computation

Let $N = (n_{i,j})_{i,j} \in \mathcal{Y}_n$. Fix $1 \leqslant i, j \leqslant n$ and suppose that $n_{i,j} \in ]0,1[$.
Prove that by replacing $n_{i,j}$ either by 0 or by 1, we can obtain a matrix $N'$ of $\mathcal{Y}_n$ such that $\operatorname{det}(N) \leqslant \operatorname{det}(N')$.
Deduce that $x_n = y_n$.

Let $N = (n_{i,j})_{i,j} \in \mathcal{Y}_n$. Fix $1 \leqslant i, j \leqslant n$ and suppose that $n_{i,j} \in ]0,1[$.

Prove that by replacing $n_{i,j}$ either by 0 or by 1, we can obtain a matrix $N'$ of $\mathcal{Y}_n$ such that $\operatorname{det}(N) \leqslant \operatorname{det}(N')$.

Deduce that $x_n = y_n$.

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.B.1 QI.B.2 QII.A.1 QII.A.2 QII.A.3 QII.B.1 QII.B.2 QII.B.3 QII.B.4 QIII.A.1 QIII.A.2 QIII.A.3 QIII.B.1 QIII.B.2 QIII.B.3 QIII.B.4 QIII.C QIV.A.1 QIV.A.2 QIV.A.3 QIV.A.4 QIV.A.5 QIV.A.6 QIV.A.7 QIV.B.1 QIV.B.2 QIV.B.3 QIV.B.4 QIV.B.5 QIV.B.6