Linear Independence of Power Vectors

Question

Let $p_{1},p_{2},\cdots,p_{K}$ be a non-uniform probability distribution (i.e., not all $p_{k}$s are the same) with $K$ being a fixed positive integer. Consider the following $K\times K$ matrix. $$ A=\left( \begin{array}{cccc} 1&1&\cdots& 1 \\ p_{1}&p_{2}&\cdots & p_{K} \\ p_{1}^2&p_{2}^2&\cdots &p_{K}^2 \\ \vdots& \vdots & \vdots & \vdots \\ p_{1}^{K-1}&p_{2}^{-1}K&\cdots &p_{K}^{K-1} \end{array} \right) $$ Can $\det(A)$ be zero, in other words, can the the column vectors (or row vectors) of $A$ be linearly dependent?