How do you express that each value of a matrix should be non-zero expect one position?

Question

Something like ∀x,y ≤ m, M[x, y] ≠ 0 but M[m, 1] = 0. (M is an m*m-matrix)

How should i write this with correct mathematical notation?

Answer 1

If $M=(a_{ij})_{1\leqslant i\leqslant m,1\leqslant j\leqslant n}$, I would write$$\bigl(\forall(k,l)\in(\{1,2,\ldots,n\}\times\{1,2,\ldots,n\})\setminus\{(i,j)\}\bigr):a_{kl}\neq0.$$So, $a_{ij}$ can be $0$, but all other entries of the matrix must be different from $0$.

Answer 2

I would say: "$M$ has exactly one entry equal to zero". Much easier to read and understand than a sea of symbols.