Why the addition of the noise will almost certainly make the square matrix invertible?
The matrix is invertible if its determinant is not equal to $0$. The determinant of the matrix geometrically can be understood as a $n$-dimensional parallelepiped defined by its row/column
vectors.
Now let our non-invertible matrix $A$ be for simplicity $3 \times 3$ matrix. Obviously, $det(A) = 0$ i.e. the parallelepiped formed by the rows/columns of $A$ is flat. If we add some noise matrix $V$ where $v_{ij}$~$Norm(0,a)$ to each entry of $A$ i.e. $A+V$, we will change the the rows/columns of $A$. How can one show geometrically (or any way) that the new rows/columns would be independent?