Let $x_1, x_2, x_3, x_4, y_1, y_2, y_3$ and $y_4$ be fixed real numbers, not all of them equal to zero. Define a $4 \times 4$ matrix A by
$$\left( \begin{array}{ccc} x_1^2+y_1^2 & x_1x_2+y_1y_2& x_1x_3 +y_1y_3& x_1x_4 +y_1y_4\\ x_2x_1+y_2y_1 & x_2^2+y_2^2 & x_2x_3 +y_2y_3 & x_2x_4 +y_2y_4 \\ x_3x_1+y_3y_1 & x_3x_2 +y_3y_2 & x_3^2+y_3^2 & x_3x_4 +y_3y_4 \\ x_4x_1+y_4y_1 & x_4x_2 +y_4y_2 & x_4x_3 +y_4y_3 & x_4^2+y_4^2 \end{array} \right)$$. We need to calculate its rank.
In the worst case lets assume all except one variables are zero. Then exactly one entry of the matrix is non-zero, thus the rank is 1. Is this correct ?