Probability that a $3\times 3$ matrix with entries in $\{0,1,2,3\}$ is invertible. Let $A$ be a $3\times 3$ matrix, and each of its entries takes value from $\{0, 1, 2, 3\}$ with probability $1/4$ for each value. What is the probability that A is invertible?
I have tried to list all possible combinations, but that is too complicated and I cannot proceed. Any help is appreciated.
The matrix operation is in R, not restricted modulo 4. For example, 2*2 = 4, not 0, so a matrix with determinant 4 is regarded as invertible.

Hint. For $A$ to be invertible, each column of $A$ must be linearly independent.  With the exception of all $0$'s, all vectors in $\{0,1,2,3\}$ work for the first column, so there are $4^3-1$ possibilities for this vector.  Then, for the second column, you can choose any vector which is linearly independent from the first column vector.  Can you count the possible ways this can happen?  The third column should be done similarly.
