I'm working with a square matrix $A$ that has $(2r-1)m$ rows with $m$ being an even number. Among all rows of $A$, there are $m$ rows that contain only randomly generated non-zero elements. The other $(2r-2)m$ rows contain half non-zero elements where each half-zero row has a compliment row with the position of the zero and non-zero elements swapped. All non-zero elements are randomly generated.
Now I want to show that the matrix $A$ is invertible with high probability as the field size (in which the random elements are generated) goes to infinity.
I can re-arranging the rows of $A$ to make all elements along the diagonal are non-zero. But what's the next step? Maybe the Schwartz-Zippel Lemma is useful when proving the determinant of $A$ is non-zero. Can anyone help me?