As per the definition given for $P,$ row $k$ is $k$ multiplied by row number $1.$ i.e., all other rows are linearly dependent on row number $1.$ So, rank of $P$ is $1.$
For example, let $n = 3$
$P = \begin{bmatrix}
p_{11} & p_{12} & p_{13} \\
p_{21} & p_{22} & p_{23} \\
p_{31} & p_{32} & p_{33} \\
\end{bmatrix}_{3 \times 3}$
$P = \begin{bmatrix}
1 & 2 & 3 \\
2 & 4 & 6 \\
3 & 6 & 9 \\
\end{bmatrix}_{3 \times 3}$
Now, perform the operation, $R_{2} \rightarrow R_{2} - 2R_{1}\;\text{and}\; R_{3} \rightarrow R_{3} - 3R_{1}$
$P = \begin{bmatrix}
1 & 2 & 3 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
\end{bmatrix}_{3 \times 3}$
$\therefore$ The rank of the matrix is $1.$
So, the correct answer is $1.$