JEE Advanced 2012 Paper 2
Exam JEE Advanced
Year 2012
Paper Paper 2
Subject Mathematics
Chapter Matrices JEE Advanced 2012 · Paper 2 · Mathematics · Matrices
If P is a 3 × 3 matrix such that P^ T =2 P+I , where P^ T is the...
If
P P P is a
3 × 3 3 \times 3 3 × 3 matrix such that
P T = 2 P + I P^{T}=2 P+I P T = 2 P + I , where
P T i s t h e t r a n s p o s e o f P^{T} is the transpose of P T i s t h e t r an s p oseo f P
a n d and an d I
i s t h e is the i s t h e 3 \times 3
i d e n t i t y m a t r i x , t h e n t h e r e e x i s t s a c o l u m n m a t r i x identity matrix, then there exists a column matrix i d e n t i t y ma t r i x , t h e n t h eree x i s t s a co l u mnma t r i x \quad X=
\left[ x y z \begin{array}{l}x \\ y \\ z\end{array} x y z \right] \neq\left[ 0 0 0 \begin{array}{l}0 \\ 0 \\ 0\end{array} 0 0 0 \right] such that
Options A P X=\left[ 0 0 0 \begin{array}{l}0 \\ 0 \\ 0\end{array} 0 0 0 \right] B P X = X P X=X PX = X C P X = 2 X P X=2 X PX = 2 X D P X = − X P X=-X PX = − X Correct answer D. P X = − X P X=-X PX = − X
Step-by-step solution
P T = 2 P + I P^{T}=2 P+I P T = 2 P + I ⇒ P = 2 P T + I ⇒ P = 2 ( 2 P + I ) + I ⇒ P = 4 P + 3 I ⇒ P + I = 0 ⇒ P X + X = 0 ⇒ P X = − X \begin{array}{l}
\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
\end{array} ⇒ P = 2 P T + I ⇒ P = 2 ( 2 P + I ) + I ⇒ P = 4 P + 3 I ⇒ P + I = 0 ⇒ PX + X = 0 ⇒ PX = − X
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