The correct option is: Trace of P is equal to the sum of the Eigen values of P.

Step-by-Step Explanation:

From the provided image, the matrix P is given as:

$$P=\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 0\\ 1& 0& 1\end{array}\right]$$

Let us analyze the trace of matrix P and check the validity of each statement.

1. Calculation of the Trace of P:
The trace of a matrix is the sum of its main diagonal elements. For the given 3 × 3 matrix P:
$$\text{Trace}\left(P\right)=P_{11}+P_{22}+P_{33}$$
Substituting the diagonal values from matrix P (which are 1, 1, and 1):
$$\text{Trace}\left(P\right)=1+1+1=3$$

2. Finding the Eigenvalues of P:
The eigenvalues of a matrix are determined by solving its characteristic equation, given by:
$$det(P-\lambda I)=0$$
Substituting the matrix P:
$$det\left[\begin{array}{ccc}1-\lambda & 0& 1\\ 0& 1-\lambda & 0\\ 1& 0& 1-\lambda \end{array}\right]=0$$
Expanding this determinant along the second row gives:
$$(1-\lambda )\left[\right(1-\lambda \left)\right(1-\lambda )-(1\left)\right(1\left)\right]=0$$
Simplifying the expression inside the brackets:
$$(1-\lambda )\left[\right(1-2\lambda +{\lambda }^2)-1]=0$$
$$(1-\lambda )({\lambda }^2-2\lambda )=0$$
Factoring out $\lambda$ from the second term:
$$\lambda (1-\lambda )(\lambda -2)=0$$
Thus, the eigenvalues ($\lambda$) are:
$${\lambda }_1=0,{\lambda }_2=1,{\lambda }_3=2$$

3. Sum of the Eigenvalues:
Calculating the sum of the eigenvalues:
$$\sum {\lambda }_i=0+1+2=3$$
Since $\text{Trace}\left(P\right)=3$ and the sum of the eigenvalues is also 3, we verify that the trace is indeed equal to the sum of the eigenvalues of P.

4. Verification of the Other Options:
• P^TP is an identity matrix: Since P is symmetric, $P^T=P$. Thus:
$$P^{T}P=P^2=\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 0\\ 1& 0& 1\end{array}\right]\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 0\\ 1& 0& 1\end{array}\right]=\left[\begin{array}{ccc}2& 0& 2\\ 0& 1& 0\\ 2& 0& 2\end{array}\right]\ne I$$ This statement is false.
• P is a skew-symmetric matrix: A matrix is skew-symmetric if $P^T=-P$. Here, $P^T=P$, so it is symmetric. This statement is false.
• Absolute magnitude of each Eigen value is 1: The absolute magnitudes of the eigenvalues are |0| = 0, |1| = 1, and |2| = 2. Since they are not all equal to 1, this statement is false.