If P is a 3 × 3 matrix such that P^T = 2P + I, where P^T is… — JEE Main 2012 Mathematics PYQ

AIEEE 2012

P

is a

3 \times 3

matrix such that

P^T = 2P + I

, where

P^T

is the transpose of

P

and

I

is the

3 \times 3

identity matrix, then there exists a column matrix,

X = \begin{bmatrix} x \\ y \\ z \end{bmatrix} \neq \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}

such that

A.
$PX = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$
B.
$PX = X$
C.
$PX = 2X$
D.
$PX = -X$

Show answer and solution

Correct option: D

Solution

From P^T = 2P + I, take the transpose of both sides:

P = 2P^T + I

Substitute P^T = 2P + I into this:

P = 2(2P + I) + I = 4P + 3I

So -3P = 3I, giving P = -I.

Then PX = -IX = -X for any nonzero column vector X.

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