Jee main 2022

Question

If P is a 3×3 matrix such that PT=2P+I, where PT' is the transpose of P and I is the 3×3 identity matrix, then there exists a column matrix X=⎡⎢⎣xyz⎤⎥⎦≠⎡⎢⎣000⎤⎥⎦ such that ______

Answer 1

Given that P^T = 2 P + I 

⇒ ( P T )^T = ( 2 P + I )^T 

⇒ P = 2 P^T + I 

⇒ P = 2 ( 2 P + I ) + I 

⇒ 3 P = − 3 I 

⇒ P = − I 

∴ P X = − I X 

= − X