[Math] Check if matrix determinant is zero

Question

What's the simplest way to check if a NxN Matrix determinant is zero ? Using Gauss Jordan to calculate the determinant first is to complicated (took N^3 calculation), is there any way to know it in at most (N^2 calculation).

Anyway the matrix is always in form of this (the first row is known value) :

2×2 matrix
$\begin{bmatrix}
a & b \\
b & a \\
\end{bmatrix}$

3×3 matrix
$\begin{bmatrix}
a & b & c \\
b & c & a \\
c & a & b
\end{bmatrix}$

4×4 matrix
$\begin{bmatrix}
a & b & c & d\\
b & c & d & a\\
c & d & a & b\\
d & a & b & c
\end{bmatrix}$

Answer 1

All you ever wanted to know about such matrices (including the answer to your question) can be found in this Wikipedia article about circulant matrices.