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The Invertible Matrix Theorem

Let A A be a square n×n n \times n matrix. Then the following statements are equivalent. That is, for a given A A , the statements are either all true or all false.
  1. A A is an invertible matrix.
  2. A A is row equivalent to the n×n n \times n identity matrix.
  3. A A has n n pivot positions.
  4. The equation Ax⃗=0 A\vec{x} = 0 has only the trivial solution.
  5. The columns of A A form a linearly independent set.
  6. The linear transformation x⃗↦Ax⃗ \vec{x} \mapsto A\vec{x} is one-to-one.
  7. The equation Ax⃗=b⃗ A\vec{x} = \vec{b} has at least one solution for each b⃗ \vec{b} in Rn \mathbb{R}^n .
  8. The columns of A A span Rn \mathbb{R}^n .
  9. The linear transformation x⃗↦Ax⃗ \vec{x} \mapsto A\vec{x} maps Rn \mathbb{R}^n onto Rn \mathbb{R}^n .
  10. There is an n×n n \times n matrix C C such that CA=I CA = I .
  11. There is an n×n n \times n matrix D D such that AD=I AD = I .
  12. AT A^T is an invertible matrix.

Theorem

Let T:Rn→Rn T : \mathbb{R}^n \to \mathbb{R}^n be a linear transformation and let A A be the standard matrix for T T . Then T T is invertible if and only if A A is an invertible matrix. In that case, the linear transformation S S given by S(x⃗)=A−1x⃗ S(\vec{x}) = A^{-1} \vec{x} is the unique function satisfying (1) and (2). S(T(x⃗))=x⃗for allx⃗∈Rn(1) S(T(\vec{x})) = \vec{x} \quad \text{for all} \quad \vec{x} \in \mathbb{R}^n \quad \text{(1)} T(S(x⃗))=x⃗for allx⃗∈Rn(2) T(S(\vec{x})) = \vec{x} \quad \text{for all} \quad \vec{x} \in \mathbb{R}^n \quad \text{(2)}