Here is a link to a pdf file with all 3X3 unimodular matrices with entries $-1,0,1,2,3$ whose inverses have entries among the ten digits and their opposites. I have omitted the matrices with 3 or more zeros. This pdf file is large, 5MB, 3,119 pages with 87,310 unimodular matrices, each matrix accompanied with its inverse. If you ever need a nice unimodular 3X3 matrix I hope you find it here.