grandes-ecoles 2016 Q1

grandes-ecoles · France · x-ens-maths1__mp Matrices Determinant and Rank Computation

Let $M \in M_n(\mathbb{R})$ be an invertible matrix with integer coefficients.
1a. Show that $M^{-1}$ has rational coefficients.
1b. Show the equivalence of the following propositions:
i) $M^{-1}$ has integer coefficients.
ii) $\det M$ equals 1 or $-1$.

Let $M \in M_n(\mathbb{R})$ be an invertible matrix with integer coefficients.

1a. Show that $M^{-1}$ has rational coefficients.

1b. Show the equivalence of the following propositions:

i) $M^{-1}$ has integer coefficients.

ii) $\det M$ equals 1 or $-1$.

Paper Questions

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