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Thanu Dharshani posted an Question
July 13, 2021 • 10:34 am 30 points
  • IIT JAM
  • Mathematics (MA)

Let ae m,, (c). then a is diagonalizable if and only if: a) a = 0. (b) a = i. (c) n =2. d) none of the other three options.

Let AE M,, (C). Then A is diagonalizable if and only if: a) A = 0. (b) A = I. (c) n =2. d) None of the other three options.

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  • Thanu dharshani 1

    but matrix with trace 0 is diagonalisable?

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    yes it can be possible because let zero matrix or take a diagonal non zero matrix whose trace is zero then it will be diagonalizable

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    you can take 2*2 matrix whose diagonal entries are 1 and -1 and other entries are 0 . trace of this matrix is zero and also it is diagonalizable

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    diagonalizability doesn't depend on trace it depend on AM and GM of each Eigen value. AM should be equal to GM for each Eigen value.

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    ask if you have any doubts

  • Ankur Rao best-answer

    option (a) is true only. a) as if A is zero matrix then the above matrix will also be zero and we know zero matrix is diagonal matrix contain zero in diagonals. so it will be true. c) it can be true for n=1,2,3,4,5... , so this is false b) take A={1} then it have 1 as Eigen value and arithmetic multiplicity(AM) is 2 but the geometric multiplicity(GM) is 1 , then it will be not diagonalizable because for diagonalizability AM=GM for any Eigen value , so option b) is also false

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