If P is a 3 × 3 matrix such that P T = 2 P + I , where P T is the transpose of P and I is the 3 × 3 identity matrix, then there exists a column matrix X = [ x y z ] [ 0 0 0 ] such that [2012]

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 $3 \times 3$ identity matrix, then there exists a column matrix $X = [\begin{matrix} x \\ y \\ z \end{matrix}] \neq [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$ such that [2012]

Smart Achievers · Accepted Answer

(4)

$P^{T} = 2 P + I$

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

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

$\Rightarrow P = 4 P + 3 I$

$\Rightarrow P + I = 0$

$\Rightarrow P X + X = 0$

$\Rightarrow P X = - X$

Solution

Q.
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 $3 \times 3$ identity matrix, then there exists a column matrix $X = [\begin{matrix} x \\ y \\ z \end{matrix}] \neq [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$ such that [2012]

1 $P X = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$

2 $P X = X$

3 $P X = 2 X$

4 $P X = - X$

Ans.
(4)

$P^{T} = 2 P + I$

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

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

$\Rightarrow P = 4 P + 3 I$

$\Rightarrow P + I = 0$

$\Rightarrow P X + X = 0$

$\Rightarrow P X = - X$

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Solution

Q. 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 [2012]

1 PX=[000]

2 PX=X

3 PX=2X

4 PX=-X

Ans. (4) PT=2P+I ⇒P=2PT+I ⇒P=2(2P+I)+I ⇒P=4P+3I ⇒P+I=0 ⇒PX+X=0 ⇒PX=-X