Maths. Section B1Matrices

If P is a 3 × 3 matrix such that P^T=2P + I, where P^T is the transpo…

If P is a 3 × 3 matrix such that $P^T=2P + I$, where $P^T$ is the transpose of P and I is 3 × 3 identity, then there exists a column matrix $X=\begin{bmatrix}x\\y\\z\end{bmatrix}≠\begin{bmatrix}0\\0\\0\end{bmatrix}$ such that $PX =$

Answer & explanation

Correct answer: option 4

We have,

$P^T = 2P + I$

$⇒(P^T)^T =(2P+I)^T$

$⇒P = 2P^T + I$

$⇒P=2(2P+I) + I$

$⇒P = 4P + 3I$

$⇒3P+3I=O$

$⇒P+I=O$

$⇒P=-I$

$⇒PX=-IX$

$⇒PX=-X$