Determinant of a matrix with odd diagonal and even entries I'm trying the solve the following problem linear algebra problem and I'm not sure where to begin:
Let $B$ be a square matrix with n columns and integer entries. This matrix is constructed so that all diagonal entries are odd and all other entries are even. We wish to demonstrate that $\det(B) \neq 0$.
My knee jerk is to separate $B$ into even and odd matrices $B_{even}$ and $B_{odd}$, where $B_{odd}$ will be diagonal and have a nonzero determinant. But I'm not sure where to go from there.

$$B = \begin{bmatrix} 
    \color{red}{odd} & even & \dots & even\\
    even & odd & \dots & even \\ 
    \vdots & \ddots & \dots & even\\
    even & even &even & odd 
    \end{bmatrix}$$
$$|B| = \color{red}{odd} + even+even+..........+even=odd\ne 0$$
