GATE 2020 MA – Question 38

Linear Algebra · Eigenvalues and Canonical Forms · 2 marks · Multiple choice

Let P∈Mₘ×ₙ(R). I: if XPY = 0 for every row X∈M₁×ₘ(R) and column Y∈Mₙ×₁(R), then P = 0. II: if m = n, P is symmetric and P² = 0, then P = 0.

Both I and II are true.
I is true but II is false.
I is false but II is true.
Both I and II are false.

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Correct answer: (A) Both I and II are true.

Explanation

Choose X and Y to be coordinate row and column vectors. XPY then selects any chosen entry of P, so all entries vanish and I is true. For II, symmetry gives $\|Pv\|^2=v^TP^TPv=v^TP^2v=0$ for every v, so Pv = 0 for all v and P = 0. Both statements are true, A.

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