Olympiad problems

2019 Korea USCM, 8

$M_n(\mathbb{C})$ is the vector space of all complex $n\times n$ matrices. Given a linear map $T:M_n(\mathbb{C})\to M_n(\mathbb{C})$ s.t. $\det (A)=\det(T(A))$ for every $A\in M_n(\mathbb{C})$. (1) If $T(A)$ is the zero matrix, then show that $A$ is also the zero matrix. (2) Prove that $\text{rank} (A)=\text{rank} (T(A))$ for any $A\in M_n(\mathbb{C})$.

matrix matrices linear algebra vector space

2017 Romania National Olympiad, 2

Let be two natural numbers $ n\ge 2, k, $ and $ k\quad n\times n $ symmetric real matrices $ A_1,A_2,\ldots ,A_k. $ Then, the following relations are equivalent: $ 1)\quad \left| \sum_{i=1}^k A_i^2 \right| =0 $ $ 2)\quad \left| \sum_{i=1}^k A_iB_i \right| =0,\quad\forall B_1,B_2,\ldots ,B_k\in \mathcal{M}_n\left( \mathbb{R} \right) $ $ || $ [i]denotes the determinant.[/i]

linear algebra vector space

2002 SNSB Admission, 1

Let $ u,v $ be two endomorphisms of a finite vectorial space that verify the relation $ uv-vu=u. $ Calculate $ u^kv-vu^k $ and show that u is nilpotent.

linear algebra field theory vector space mathematical induction