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Question:

If P is a 3x3 matrix such that PT = 2P + I, where PT is the transpose of P and I is the 3x3 identity matrix, then there exists a column matrix X = [[x], [y], [z]] ≠ [[0], [0], [0]] such that PX = [[0], [0], [0]]

PX=[[2], [0], [0]]

PX=3X

PX=−X

PX=−X

PX=5X

PX=[[0], [0], [0]]

PX=X

PX=2X

Solution:

Given that PT = 2P + I ⇒ (PT)T = (2P + I)T ⇒ P = 2PT + I ⇒ P = 2(2P + I) + I ⇒ P = 4P + 2I + I ⇒ 3P = -3I ⇒ P = -I
Therefore, PX = -IX = -X