TIFR 2015 Problem 3 Solution is a part of TIFR entrance preparation series. The Tata Institute of Fundamental Research is India's premier institution for advanced research in Mathematics. The Institute runs a graduate programme leading to the award of Ph.D., Integrated M.Sc.-Ph.D. as well as M.Sc. degree in certain subjects.
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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?
A. There exists a matrix B such that (AB-BA=B)
B. There exists a matrix B such that (AB-BA=A)
C. There exists a matrix B such that (AB+BA=A)
D. There exists a matrix B such that (AB+BA=B)
Discussion:
We know that for two square matrix (A) and (B) of same size, (Tr(AB)=Tr(BA)) ((TrM)=Trace of (M) ).
In other words, (Tr(AB-BA)=0) for any two square matrices of the same size.
Since trace of a square matrix is also the sum of its eigenvalues, and (A) has all eigenvalues non-negative with at least one positive eigenvalue, we have (Tr(A) > 0). Taking trace of both sides of (AB-BA=A) we get a contradiction. So there does not exist any (B) such that (AB-BA=A).
Take (B=0) the 10x10 zero-matrix. Then (AB-BA=B) is satisfied. So is (AB+BA=B).
Take (B=\frac{1}{2}I), where (I) is the 10x10 identity matrix. Then (AB+BA=A) is satisfied.
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