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Mathematics Question on Transpose of a Matrix

Let x be an n×1n \times 1 matrix. Let O and I be the zero, and identity matrices of order n, respectively. Define P=−xxTxTxP = - \frac{xx^T}{x^Tx} is the transpose of x. Then which of the following options is always CORRECT?

A

P2−P=OP^2 - P = O

B

P2−P=IP^2 - P = I

C

P2+P=OP^2 + P = O

D

P2+P=IP^2 + P = I

Answer

P2+P=OP^2 + P = O

Explanation

Solution

Since it is given that
P=−xxTxTxP= - \frac{x x^{T}}{x^{T} x}
⇒xTxP=−xxT\Rightarrow x^{T} xP = -xx^{T}
On applying transpose both sides, we get
PTxTx=−xTxP^{T}x^{T}x = -x^{T}x
⇒(PT+I)xTx=O\Rightarrow \left(P^{T} +I\right) x^{T}x = O
⇒PT+I=0\Rightarrow P^{T} + I =0
\Rightarrow P^T = -I \therefore P = -I \left\\{ \because I^{T} = I\right\\}
so P2+P=I−I=OP^{2} +P = I - I =O
⇒P2+P=O\Rightarrow P^{2} +P = O

Therefore, The Correct Answer is (C) P 2 + P = O