# Check if matrix determinant is zero

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) :

2x2 matrix $\begin{bmatrix} a & b \\ b & a \\ \end{bmatrix}$

3x3 matrix $\begin{bmatrix} a & b & c \\ b & c & a \\ c & a & b \end{bmatrix}$

4x4 matrix $\begin{bmatrix} a & b & c & d\\ b & c & d & a\\ c & d & a & b\\ d & a & b & c \end{bmatrix}$