Probability that a random integer matrix is invertible Assume that $A_{n\times m}$ is a random matrix, i.e. each of $A$'s entries is selected independently from a uniform distribution over $\mathbf{Z}$.
I want to show that $Pr(A\mbox{ is invertible})=1$.
I only know how to show this for $n=2$ because:


*

*With probability 1 all entries are not zero (probability to select uniformly a single element from infinite set is zero).

*Given 1, if we selected $A_{1,1}$, $A_{1,2}$ and $A_{2,1}$ and $\lambda \in\mathbf{Z}$ is the (only, if it exist) constant satisfying $A_{2,1} = \lambda A_{1,1}$ then if we select $A_{2,2} \neq\lambda A_{1,2}$ (which happens with probability 1 from the same considerations) then line 2 is linearly independent in line 1.


However the same argument will not work for line 3 since there are infinitely many combinations of $\lambda_1, \lambda_2$ that satisfy $A_{3,1} = \lambda_1A_{1,1}+ \lambda_2 A_{2,1}$.
 A: On any line $A+tB$ in matrix space, the set of singular matrices is given by the solutions of the polynomial equation
$$
\det(A+tB)=0
$$
Since there are only finitely many, this strongly hints that for most probability distributions the chance of hitting a singular matrix is zero.
Or told another way, the set of singular matrices forms a hypersurface, which thus has Lebesgue measure zero. Any probability measure that has a Radon-Nikodym density relative to the Lebesgue measure will also have the singular matrices as nullset.
A: A single random $2 \times 2$ matrix is invertible iff the determinant $ad-bc \neq 0$:
$$ \left[\begin{array}{cc} a & b \\ c & d \end{array}\right]$$
Regardless of what distribution we put on $\mathbb{Z}$, we can find the probability of the even $\mathbb{P}[ad-bc = 0]$ or its complement.

If we have a $3 \times 2$ matrix - I am not entirely sure what "invertible" means in this case - but we have more events to consider and they are not independent:
$$ \left[\begin{array}{ccc} a & b & c\\ d & e & f\end{array}\right]$$
We need that $ae-bd, af-cd,bf-ce\neq 0$.  In this case, one might try to study the rank of this matrix. 


*

*the rank is $0$ if all entries are $0$

*the rank is $1$ if $d = ka, e = kb, f = kc$ for some $k$.

*the rank is $2$ if the rows are not proportional


$\mathbb{P}[rank = 0]$ is very small in some sense, but rank 1 is possible if all 3 "coincidences" above do occur.

We have said all of this without specifying a measure on the set of integers $(\mathbb{Z}, \mu)$.  On possibility is to set $\mu$ to be the uniform measure on $[1,N]$ and let $N \to \infty$ but it's certainly not the only choice.
