All elements of a 100 x 100 matrix ,A are odd numbers. What is the greatest natural number that would always divide the determinant of A?
I have been able to show that it is always divisible by 2^99(y performing elementary row subtractions and additions) , but am stuck when it comes to showing that there is no natural number greater than this.
Consider the matrix,
$ M= \begin{bmatrix} 1 & 1 & 1 & 1 & \cdots & 1 \\ 1 & 3 & 1 & 1 & \cdots & 1 \\ 1 & 1 & 3 & 1 & \cdots & 1 \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots\\ \end{bmatrix} $
This gives $2^{99}$ is the greatest natural number that would always divide the determinant.