The Invertible Matrix theorem

Let A be a n x n matrix. Then, the following statements are equivalent.

a. A is an invertible matrix.
b. A is row equivalent to the n x n identity matrix.
c. A has n pivot positions.
d. The equation Ax = 0 has only the trivial solution.
e. The columns of A form a linearly independent set.
f. The linear transformation x -> Ax is one-to-one.
g. The equation Ax = b has at least one solution for each b in Rⁿ.
h. The columns of A span Rⁿ.
i. The linear transformation x -> Ax maps Rⁿ onto Rⁿ.
j. There is an n x n matrix C such that CA = I.
k. There is an n x n matrix D such that AD = I.
l. A^T is an invertible matrix.
m. The columns of A form a basis of Rⁿ.
n. Col A = Rⁿ.
o. dim Col A = n.
p. rank A = n.
q. Nul A = {0}.
r. dim Nul A = 0.
s. The number 0 is not an eigenvalue of A.
t. The determinant of A is not 0.