Let A be a square n × n matrix. Then the following are equivalent:

(i) A is an invertible matrix.

(ii) A is row equivalent to the n × n identity matrix.

(iii) A has n pivots positions.

(iv) The equation Ax = 0 has only the trivial solution.

(v) The columns of A form a linearly independent set.

(vi) The linear transformation x ↦ Ax is one-to-one.

(vii) The equation Ax = b has at least one solution for each b in Rn.

(viii) The columns of A span Rn.

(ix) The linear transformation x ↦ Ax maps Rn onto Rn.

(x) There is an n × n matrix C such that CA = I.

(xi) There is an n × n matrix D such that AD = I.

(xii) A^T is an invertible matrix.