If P is a 3 x 3 matrix such that P<sup>T</sup> = 2P + I, where P<sup>T</sup> is the transpose of P and I is the 3 x 3 identity matrix, then there exists a column matrix X = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr><mtr><mtd><mi>z</mi></mtd></mtr></mtable></mfenced></math> ≠ <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math> such that
Step-by-Step Solution
Key Concept: Given P^T = 2P + I. Taking transpose on both sides, (P^T)^T = (2P + I)^T => P = 2P^T + I. Substitute P^T = 2P + I into this equation: P = 2(2P + I) + I = 4P + 2I + I = 4P + 3I. Thus, 3P = -3I, which implies P = -I. Therefore, PX = -IX = -X.
Given P<sup>T</sup> = 2P + I. Taking transpose on both sides, (P<sup>T</sup>)<sup>T</sup> = (2P + I)<sup>T</sup>, which gives P = 2P<sup>T</sup> + I. Substituting P<sup>T</sup> = 2P + I, we get P = 2(2P + I) + I = 4P + 2I + I = 4P + 3I. This implies 3P = -3I, so P = -I. Thus, PX = -IX = -X.
Correct Answer: 4