grandes-ecoles 2017 QI.B.4

grandes-ecoles · France · centrale-maths1__official Matrices Determinant and Rank Computation

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

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

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

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