TITLE: Is the following matrix invertible?
QUESTION [150 upvotes]: $$\begin{bmatrix} 1235 &2344 &1234 &1990\\
2124 & 4123& 1990& 3026 \\
1230 &1234 &9095 &1230\\
1262 &2312& 2324 &3907 
\end{bmatrix}$$
Clearly, its determinant is not zero and, hence, the matrix is invertible. 
Is there a more elegant way to do this?
Is there a pattern among these entries?

REPLY [500 votes]: Find the determinant.  To make calculations easier, work modulo $2$! The diagonal is $1$'s, the rest are $0$'s.  The determinant is odd, and therefore non-zero.