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Romania District Olympiad 2007 - Grade XI

Source:

April 10, 2011
linear algebralinear algebra unsolved

Problem Statement

Let A∈Mn(R∗)A\in \mathcal{M}_n(\mathbb{R}^*). If A⋅ tA=InA\cdot\ ^t A=I_n, prove that:
a)∣Tr(A)∣≤n|\text{Tr}(A)|\le n;
b)If nn is odd, then det⁡(A2−In)=0\det(A^2-I_n)=0.
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