linear algebra - Determinant of the following $2018 times 2018$ matrix

Determinant of the following $2018 \times 2018$ matrix and let $B$ be the leading principal minor of $A$ of order $1009$, then rank of $B$

$$\begin{pmatrix}
0 & 2 & 0 & \ldots & \ldots & 0 \\
\frac{1}{3} & 0 & 2 & \ddots & & \vdots \\
0 & \frac{1}{3} & 0 & \ddots & \ddots & \vdots \\

\vdots & \ddots & \ddots & \ddots & \ddots & 0 \\
\vdots & & \ddots & \ddots & 0 & 2 \\
0 & \ldots & \ldots & 0 & \frac{1}{3} & 0 \\
\end{pmatrix}.$$

I tried to find out some non- zero eigenvector to get the eigenvalues but I did'nt get it. Please help on this problem.