Invertible Matrix Theorem
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.
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.