linear algebra - Find the determinant of the following matrix

Find the determinant of the following matrix:

$$A = \begin{bmatrix} 1+x_1^2 &x_1x_2 & ... & x_1x_n \\ x_2x_1&1+x_2^2 &... & x_2x_n\\ ...& ... & ... &... \\ x_nx_1& x_nx_2 &... & 1+x_n^2 \end{bmatrix}$$

I computed for the case $n=2$, and $n=3$ and guessed that $\det(A)$ should be $1+\sum_{i=1}^n x_i^2$ but not sure how to proceed for any $n$.