Is there a more efficient way to calculate the determinant of this matrix? I used Gaussian Elimination to calculate $$\det\begin{pmatrix}1&4&9&16&25&36\\4&9&16&25&36&49\\9&16&25&36&49&64\\16&25&36&49&64&81\\25&36&49&64&81&100\\36&49&64&81&100&121\end{pmatrix}$$ and found the answer to be $0$. It took a lot of time to do.
The lower-upper (LU) decomposition is shown below. I think that there might be a more efficient way to calculate the determinant of this kind of matrix.


Let us give your matrix a name:
$$
A=
\begin{pmatrix}
1&4&9&16&25&36\\4&9&16&25&36&49\\9&16&25&36&49&64\\16&25&36&49&64&81\\25&36&49&64&81&100\\36&49&64&81&100&121
\end{pmatrix}
$$
The decomposition $A = LU$ (which you have found explicitly) is very helpful in this case. The upper right matrix ($U$) does not have full (row) rank, thus its determinant is $0$. Since the determinant is a multiplicative function, we get
$$
\det A = \det(L) \cdot \det(U) = \det(L) \cdot 0 = 0
$$
