In this problem you will be asked to apply the Invertible Matrix Theorem, which is stated here for your convenience.

Invertible Matrix Theorem
For an $n \times n$ matrix $A$, the following statements are equivalent.

a) $A$ is an invertible matrix.
b) $A\sim I_n$.
c) $A$ has $n$ pivot positions.
d) The equation $A\mathbf{x} = \mathbf{0}$ has only the trivial solution.
e) The columns of $A$ are linearly independent.
f) The linear transformation $\mathbf{x} \mapsto A\mathbf{x}$ is one-to-one.
g) The equation $A\mathbf{x} = \mathbf{b}$ has at least one solution for each $\mathbf{b}\in\mathbb{R}^n$.
h) The columns of $A$ span $\mathbb{R}^n$.
i) The linear transformation $\mathbf{x} \mapsto A\mathbf{x}$ maps $\mathbb{R}^n$ onto $\mathbb{R}^n$.
j) There is an $n\times n$ matrix $C$ such that $CA = I_n$.
k) There is an $n\times n$ matrix $D$ such that $AD = I_n$.
l) $A^T$ is an invertible matrix.

Is this statement true or false?
: If a square matrix has two identical columns, then the matrix is invertible.

Choose the implication that most directly proves the truth or falsehood of the statement above. Be sure to choose an implication in the same direction and meaning as the associated true statement, where the left side of this "if-then" statement represents the premise of the statement above.
If is , then is .

You can earn partial credit on this problem.