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 $n$.