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B6

Part of 1985 Putnam

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Putnam 1985 B6

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8/5/2019
Let GG be a finite set of real n×nn \times n matrices {Mi},1≤i≤r,\left\{M_{i}\right\}, 1 \leq i \leq r, which form a group under matrix multiplication. Suppose that ∑i=1rtr⁡(Mi)=0,\textstyle\sum_{i=1}^{r} \operatorname{tr}\left(M_{i}\right)=0, where tr⁡(A)\operatorname{tr}(A) denotes the trace of the matrix A.A . Prove that ∑i=1rMi\textstyle\sum_{i=1}^{r} M_{i} is the n×nn \times n zero matrix.
Putnamlinear algebra