JEE Advance - Mathematics (2012 - Paper 2 Offline - No. 16)
If P is a 3 $$\times$$ 3 matrix such that PT = 2P + I, where PT is the transpose of P and I is the 3 $$\times$$ 3 identity matrix, then there exists a column matrix $$X = \left[ {\matrix{
x \cr
y \cr
z \cr
} } \right] \ne \left[ {\matrix{
0 \cr
0 \cr
0 \cr
} } \right]$$ such that
$$PX = \left[ {\matrix{
0 \cr
0 \cr
0 \cr
} } \right]$$
PX = X
PX = 2X
PX = $$-$$X
Explanation
We have $${P^T} = 2P + I$$
We get $${P^T} - 2P = I$$ ..... (i)
Taking transpose, we have $${({P^T} - 2P)^T} = {I^T}$$
$$ \Rightarrow P - 2{P^T} = I$$ ..... (ii)
From (i) and (ii) on eliminating PT we have
$$ - 4P + P = 3I \Rightarrow P = - I$$ $$\therefore$$ $$P + I = 0$$
Thus, $$(P + 1)X = 0 \Rightarrow PX = - X$$
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