GATE 2020 ST – Question 11

Matrix Theory · Eigenvalues and Matrix Decompositions · 1 mark · Multiple choice

M is a nonzero3×3 idempotent matrix and I the identity. Which statement is FALSE?

Its eigenvalues can only be0 or1.
Rank(M)=Trace(M).
I−M is idempotent.
(I+M)⁻¹=I−2M.

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Correct answer: (D) (I+M)⁻¹=I−2M.

Explanation

Because M²=M, multiplying (I+M)(I−M/2) gives I. The inverse is therefore I−M/2, so D is false. Idempotence also implies eigenvalues0 or1, rank equal to trace, and (I−M)²=I−M.

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