Question

If $P$ is a $3\times 3$matrix such that ${\mathrm{P}}^{\mathrm{T}}=2\mathrm{P}+\mathrm{I}$ , where ${\mathrm{P}}^{\mathrm{T}}$ is the transpose of $P$ and I is the $3\times 3$ identity matrix, then there exists a column matrix,

$x=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]\ne \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$

such that

• A. $\mathrm{PX}=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$
• B. $\mathrm{PX}$
• C. $\mathrm{PX}=2\mathrm{X}$
• D. $\mathrm{PX}=-\mathrm{X}$

Answer 1

The correct option is D

$\mathrm{PX}=-\mathrm{X}$

Explanation for the correct options:

Given, ${\mathrm{P}}^{\mathrm{T}}=2\mathrm{P}+\mathrm{I}$

Step 1: Apply matrix rule

$\Rightarrow$ ${\left({\mathrm{P}}^{\mathrm{T}}\right)}^{\mathrm{T}}={(2\mathrm{P}+\mathrm{I})}^{\mathrm{T}}$

$\Rightarrow$$\left(\mathrm{P}\right)=2{\mathrm{P}}^{\mathrm{T}}+\mathrm{I}$

Step 2 : Put Value of ${\mathrm{P}}^{\mathrm{T}}$as $2\mathrm{P}+\mathrm{I}$

$\Rightarrow$ $\mathrm{P}=2(2\mathrm{P}+\mathrm{I})+\mathrm{I}$

$\Rightarrow$$\mathrm{P}=4\mathrm{P}+3\mathrm{I}$

$\Rightarrow$ $\mathrm{P}-4\mathrm{P}=3\mathrm{I}$

$\Rightarrow$ $-3\mathrm{P}=3\mathrm{I}$

$\Rightarrow$ $-\mathrm{P}=\mathrm{I}$

$\Rightarrow$ $\mathrm{P}=-\mathrm{I}$

$\Rightarrow$$\mathrm{PX}=-\mathrm{X}$

Hence, option ‘C’ is correct.