The following matrix is the result of a special kind of balanced signed graph of order $n$. In the Matrix $n_1,n_2,..,n_k$ are positive integers, which satisfy $\sum n_i=n.$ Prove that this matrix has two zero eigenvalues if and only if $k=6r$ for any positive integer $r$.

\begin{equation*} T_\lambda=\begin{bmatrix} -n_1 & n_2 & 0 &.&.&0 & n_k \\ n_1& -n_2 &n_3 &0&.&. & 0 \\ 0& n_2 & -n_3 & n_4 & 0& . & 0 \\ .&0 & n_3 &-n_4 &n_5&0&. \\ .&. &. & . &.&.&. \\ .&. &. &. &.&.&. \\ 0& 0 &. &. & n_{k-2}& -n_{k-1}& n_{k} \\ n_1 & 0 & .&.& 0& n_{k-1}& -n_k \end{bmatrix} \end{equation*}

I have calculated row reduced echelon forms of the above matrix by the Mathematica. It has two zero eigenvalues for $k = 6r$. I would like to have an analytic proof.

Thank you for your help in advance.