TIFR-2015-Maths-A-3

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Let $A$ be a $10 \times 10$ matrix with complex entries such that all its eigenvalues are non-negative real numbers, and at least one eigenvalue is positive. Which of the following statements is always false ?

1. There exists a matrix $B$ such that $AB-BA = B$.
2. There exists a matrix $B$ such that $AB-BA = A$.
3. There exists a matrix $B$ such that $AB+BA=A$.
4. There exists a matrix $B$ such that $AB+BA=B$.

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