Consider the matrix          

The number of distinct eigenvalues of P is

The correct answer is 1.

To find the number of distinct eigenvalues of the matrix $P$, we first identify the matrix from the given image:

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

We observe that $P$ is an upper triangular matrix, because all entries below the main diagonal are zero.
A fundamental property of upper triangular (and lower triangular) matrices is that their eigenvalues are exactly the elements located on their main diagonal.

The main diagonal elements of matrix $P$ are:
$d_{11}=1$,
$d_{22}=1$, and
$d_{33}=1$.

Thus, the eigenvalues of matrix $P$ are:
${\lambda }_1=1$, ${\lambda }_2=1$, and ${\lambda }_3=1$.

Since all the eigenvalues are equal to 1, the number of distinct eigenvalues of $P$ is 1.