I was compiling a list of non-singular equivalencies. This is all I have so far. I would appreciate if you can help me to add more to these. Let A be a square n x n matrix. The following statements are equivalent. That is, for a given A, the statements are either all true or all false. 1. A is non-singular. 2. A is row equivalent to In. 3. Ax=0 has only the trivial solution. 4. Ax=b has a unique solution, for each vector b in Rn . 5. Ax=b has at least one solution, for each vector b in Rn . 6. det(A) ≠ 0 7. The column vectors of A form a linearly independent set in Rn. 8. A(transpose) is non-singular. 9. The column vectors of A span Rn. 10. The column vectors of A form a basis for Rn. I am thinking in the range of topics like basis, dimension, rank, column space, row space, null space, etc. any help is highly appreciated.