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 identify matrix- then there exists a column matrix X- x- y- z - 0- 0- 0 - such that-

The correct option is C PX = X P^T=2P+I ⇒ (P^T)^T=(2p+I)^T=2P^T+I ⇒ P=2P^T+I (∵ (P^T)^T=P) ⇒ P=2(2p+I)+I ⇒ 3P= 3I⇒ P= I ...

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Standard X

Mathematics

Diagonal Matrix

If P is a 3...

Question

If P is a $3 \times 3$ matrix such that $P^{T} = 2 P + I$ where $P^{T}$ is the transpose of P and I is the $3 \times 3$ identify matrix, then there exists a column matrix $X = ⎡ ⎢ ⎣ \begin{matrix} x y z \end{matrix} ⎤ ⎥ ⎦ \neq ⎡ ⎢ ⎣ \begin{matrix} 000 \end{matrix} ⎤ ⎥ ⎦$ such that.

P X = ⎡ ⎢ ⎣ \begin{matrix} 000 \end{matrix} ⎤ ⎥ ⎦

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= -

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Solution

The correct option is C PX $= -$ X
$P^{T} = 2 P + I$

$\Rightarrow (P^{T})^{T} = (2 p + I)^{T} = 2 P^{T} + I$

$\Rightarrow P = 2 P^{T} + I$ $(∵ (P^{T})^{T} = P)$

$\Rightarrow P = 2 (2 p + I) + I$

$\Rightarrow 3 P = - 3 I \Rightarrow P = - I$

$\Rightarrow P X = - I X = - X$ .

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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 identify matrix, then there exists a column matrix X=⎡⎢⎣xyz⎤⎥⎦≠⎡⎢⎣000⎤⎥⎦ such that.

The correct option is C PX =−XPT=2P+I⇒(PT)T=(2p+I)T=2PT+I⇒P=2PT+I (∵(PT)T=P)⇒P=2(2p+I)+I⇒3P=−3I⇒P=−I⇒PX=−IX=−X.

The correct option is C PX $= -$ X
$P^{T} = 2 P + I$

$\Rightarrow (P^{T})^{T} = (2 p + I)^{T} = 2 P^{T} + I$

$\Rightarrow P = 2 P^{T} + I$ $(∵ (P^{T})^{T} = P)$

$\Rightarrow P = 2 (2 p + I) + I$

$\Rightarrow 3 P = - 3 I \Rightarrow P = - I$

$\Rightarrow P X = - I X = - X$ .