Easier way of calculating the determinant for this matrix I have to calculate the determinant of this matrix:
$$
\begin{pmatrix}
a&b&c&d\\b&c&d&a\\c&d&a&b\\d&a&b&c
\end{pmatrix}
$$
Is there an easier way of calculating this rather than the long regular way?

You can easily transform your matrix to a circulant matrix
$$
M=\left(\begin{array}{cccc}
a&b&c&d\\
d&a&b&c\\
c&d&a&b\\
b&c&d&a
\end{array}\right)
$$
by carrying out obvious row swaps. The eigenvalue theory of circulant matrices is completely known. In the $4\times 4$ case the eigenvalues of $M$ are
$$
\lambda_1=a+b+c+d,\ \lambda_2=a+bi-c-di,\ \lambda_3=a-b+c-d\ \text{and}\ \lambda_4=a-bi-c+di.
$$
The determinant of a matrix is the product of its eigenvalues so
$$
\det M=\lambda_1\lambda_2\lambda_3\lambda_4.
$$
