grandes-ecoles 2017 QI.B.4

grandes-ecoles · France · centrale-maths1__mp Matrices Determinant and Rank Computation

We consider $A \in \mathcal{M}_{n}(\mathbb{R})$. We assume $A$ is invertible and, in accordance with the notations of the problem, $(A^{-1})_{s}$ denotes the symmetric part of the inverse of $A$. Show that $(\operatorname{det}(A))^{2}\operatorname{det}\left((A^{-1})_{s}\right) = \operatorname{det}(A_{s})$.
One may consider $A(A^{-1})_{s}A^{\top}$.

We consider $A \in \mathcal{M}_{n}(\mathbb{R})$. We assume $A$ is invertible and, in accordance with the notations of the problem, $(A^{-1})_{s}$ denotes the symmetric part of the inverse of $A$. Show that $(\operatorname{det}(A))^{2}\operatorname{det}\left((A^{-1})_{s}\right) = \operatorname{det}(A_{s})$.

One may consider $A(A^{-1})_{s}A^{\top}$.

Paper Questions

QI.A.1 QI.A.2 QI.B.1 QI.B.2 QI.B.3 QI.B.4 QI.C.1 QI.C.2 QI.C.3 QII.A.1 QII.A.2 QII.A.3 QII.A.4 QII.A.5 QII.A.6 QII.A.7 QII.A.8 QII.B.1 QII.B.2 QII.B.3 QII.C.1 QII.C.2 QII.C.3 QII.C.4 QII.C.5 QII.C.6 QII.C.7 QII.C.8 QII.D.1 QII.D.2 QII.E.1 QII.E.2 QII.E.3 QII.E.4 QII.E.5 QIII.A.1 QIII.A.2 QIII.A.3 QIII.A.4 QIII.B.1 QIII.B.2 QIII.B.3 QIII.C.1 QIII.C.2 QIII.C.3