Characterizations of Invertible Matrices and the Invertible Matrix Theorem

The fundamental characterization of an $n \times n$ matrix $A$ centers on its pivot positions. These pivots determine the behavior of the matrix in linear systems and transformations.
For any square matrix $A$ of size $n \times n$ , each of the following statements is true if and only if a specific pivot condition is met:
Pivot in Every Column of $A$ :
- The equation $A\mathbf{x} = \mathbf{0}$ has only the trivial solution.
- The columns of $A$ are linearly independent.
- The linear transformation $T(\mathbf{x}) = A\mathbf{x}$ is one-to-one.
Pivot in Every Row of $A$ :
- The equation $A\mathbf{x} = \mathbf{b}$ has a solution for every $\mathbf{b}$ in $\mathbb{R}^n$ .
- The columns of $A$ span $\mathbb{R}^n$ .
- The linear transformation $T(\mathbf{x}) = A\mathbf{x}$ is onto.
Special Case for Square Matrices ( $m = n$ ):
- For a square matrix, having a pivot in every column is logically equivalent to having a pivot in every row. If one of these conditions holds, both must hold.

The Invertible Matrix Theorem provides a comprehensive list of properties that are logically equivalent for an $n \times n$ matrix $A$ . If any one of these statements is true for a particular matrix $A$ , then they are all true. Conversely, if one is false, they are all false.
The Equivalent Statements:
- (a) The matrix $A$ is an invertible matrix.
- (b) The matrix $A$ is row equivalent to the $n \times n$ identity matrix $I_n$ .
- (c) The matrix $A$ has $n$ pivot positions.
- (d) The equation $A\mathbf{x} = \mathbf{0}$ has only the trivial solution.
- (e) The columns of $A$ form a linearly independent set.
- (f) The linear transformation $T(\mathbf{x}) = A\mathbf{x}$ 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 $T(\mathbf{x}) = A\mathbf{x}$ maps onto $\mathbb{R}^n$ .
- (j) There exists an $n \times n$ matrix $C$ such that $CA = I$ .
- (k) There exists an $n \times n$ matrix $D$ such that $AD = I$ .
- (l) The transpose matrix $A^T$ is an invertible matrix.

The proof of the Invertible Matrix Theorem typically follows a "circle of implications" where several statements are linked in a chain (e.g., $(a) \implies (b) \implies (c) \dots \implies (a)$ ).
Specific Logical Links:
- Statement (k) is linked because the inverse matrix itself works as the matrix $D$ .
- Statements (g), (h), and (i) are known to be equivalent for any matrix, connecting them as a block to the theorem.
- Statements (d), (e), and (f) are also equivalent for any matrix $A$ , connecting them as a separate block.
- The transpose $(A^T)$ link is established by the property that if $A$ is invertible, then $(A^T)^{-1} = (A^{-1})^T$ .

The Invertible Matrix Theorem divides the set of all possible $n \times n$ matrices into two distinct, disjoint classes:
- Invertible (Nonsingular) Matrices: These meet all the criteria listed in the IMT.
- Noninvertible (Singular) Matrices: These fail all the criteria listed in the IMT.
Properties of Singular Matrices: A singular $n \times n$ matrix will:
- Not be row equivalent to the identity matrix $I_n$ .
- Have fewer than $n$ pivot positions.
- Have columns that are linearly dependent.
- Fail to span $\mathbb{R}^n$ .
If a matrix $A$ is invertible, it is also true that its inverse $A^{-1}$ is invertible, with $(A^{-1})^{-1} = A$ .

Example Application: To determine if a matrix is invertible, one can perform row reduction. If a $3 \times 3$ matrix is reduced and found to have three pivot positions, it is invertible by statement (c) of the IMT.
Constraint - Square Matrices Only: The Invertible Matrix Theorem applies exclusively to square matrices.
- For example, if a $4 \times 3$ matrix (non-square) has linearly independent columns, the IMT cannot be used to make conclusions about the existence of solutions for $A\mathbf{x} = \mathbf{b}$ . The theorem does not bridge column independence and the existence of solutions for non-square matrices.

Matrix multiplication corresponds directly to the composition of linear transformations. When matrix $A$ is invertible, it describes a transformation that can be "undone."
Definition of an Invertible Transformation: A linear transformation $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is invertible if there exists a function $S: \mathbb{R}^n \rightarrow \mathbb{R}^n$ such that:
1. $S(T(\mathbf{x})) = \mathbf{x}$ for all $\mathbf{x}$ in $\mathbb{R}^n$ .
2. $T(S(\mathbf{x})) = \mathbf{x}$ for all $\mathbf{x}$ in $\mathbb{R}^n$ .
Theorem 9: Let $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a linear transformation and let $A$ be its standard matrix. $T$ is invertible if and only if $A$ is an invertible matrix.
In this case, the inverse transformation $S$ is unique and is represented by the matrix $A^{-1}$ , where $S(\mathbf{x}) = A^{-1}\mathbf{x}$ .

Part 1: If $T$ is invertible, then $A$ is invertible:
- Assume $T$ is invertible. By the definition of an inverse function, $T$ must map onto $\mathbb{R}^n$ .
- Proof: For any $\mathbf{b}$ in $\mathbb{R}^n$ , let $\mathbf{x} = S(\mathbf{b})$ . Then $T(\mathbf{x}) = T(S(\mathbf{b})) = \mathbf{b}$ , showing every $\mathbf{b}$ is in the range.
- Since $T$ is onto, statement (i) of the Invertible Matrix Theorem implies matrix $A$ is invertible.
Part 2: If $A$ is invertible, then $T$ is invertible:
- Conversely, if $A$ is invertible, we define a transformation $S(\mathbf{x}) = A^{-1}\mathbf{x}$ .
- By the properties of matrix inverses: $S(T(\mathbf{x})) = A^{-1}(A\mathbf{x}) = I\mathbf{x} = \mathbf{x}$ .
- Similarly: $T(S(\mathbf{x})) = A(A^{-1}\mathbf{x}) = I\mathbf{x} = \mathbf{x}$ .
- This satisfies both conditions of an invertible transformation.