Which statement is false ?(Linear algebra problem)

Question

Let $P=\dfrac{xx^{T}}{x^{T}x}$ be an a square matrix of order n where $x$ is a non zero column vector. Then which one of the following statement is False.

$(A)$ P is idempotent

$(B)$ P is orthogonal

$(C)$ P is symmetric

$(D)$ Rank of P is one

In this question i only know that rank of $xx^{T}$ is $1$. And some short notes i have in my mind are :

Eigen values of idempotent matrix are $0$ and $1$.

Eigen values of orthogonal matrix are $-1$ and $1$.

Eigen values of symmetric matrix are Real. But that is not sufficient for this problem i guess. I don't know how to deal rational functions in linear algebra. Please give me some knowledge i just started learning algebra.

Answer 1

Since it is idempotent it can't be orthogonal because:$$PP^T=P^TP=\dfrac{xx^Txx^T}{x^Txx^Tx}=\dfrac{x^Txxx^T}{x^Txx^Tx}=\dfrac{xx^T}{x^Tx}=P\ne I$$

Answer 2

A matrix of rank $1 $ cannot be orthogonal. Orthogonal matrices have maximal rank.