Get ready Part 1

Algebra Level 4

Let $G$ be a finite set of real $n \times n$ matrices $(M_i),1 \le i \le r,$ which form a group under matrix multiplication.Suppose that $\displaystyle \sum_{i=1}^r (tr(M_i))=0$ ,where $tr(A)$ denotes the trace of the matrix $A$ .Is $\displaystyle \sum_{i=1}^r (M_i)$ the $n \times n$ zero matrix?

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1 solution

Otto Bretscher
Oct 27, 2018

Let $M=\sum_{i=1}^rM_i$ . Then $M_i M=M$ for all $i$ , so, $M^2=rM$ and $M(M-rI_n)=0$ . Thus $M$ is diagonalisable and $0,r$ are the only possible eigenvalues. Since $tr(M)=r\times rank (M)=0$ , we have $rank (M)=0$ and $M =0$ .

Get ready Part 1

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