Consider the following matrix \begin{align} \begin{pmatrix} p_1(1-p_1) & -p_1p_2 & \cdots & -p_1p_k \\ \vdots & \vdots & \ddots & \vdots \\ -p_kp_1 & -p_kp_2 & \cdots & -p_k(1-p_k) \end{pmatrix} \end{align} with $\sum_{i=1}^k p_i = 1$. Is this matrix singular? I think it is, and I'm sure that it is for $k=1,2$ and also for simple cases if $k=3$. At deciding whether a matrix is invertible or not, the only theorem I remember is that only singular matrices have determinant zero. Unfortunately the diagonal entries make it not so easy to write this matrix in another way thereby not changing the determinant.
Is there anybody with an idea? I thank in advance, because I can't do that in the comments and of course because I'm grateful.