Given square matrix $A_{k\times k}$ with $k\geq4$. The entry of the matrix $A$ is $1, 2, \ldots, k^2$ and let $n(A)$ is the number of invertible matrix $A$. Find infimum of $n(A)$.
First, if $k$ increase, we must show that infimum $n(A)$ will increase. So infimum of $n(A)$ will attain when $k=4$.
How we compute the number of this Invertible matrix? Is compute the number of matrix with rank 2 or 3 necessary?