Matrix Inverse

Definition

$A^{-1}$ is the inverse of $A$ if:

$A^{-1}A = AA^{-1} = I$

Existence: Inverse exists if and only if $\det(A) \neq 0$.

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Methods for Finding Inverse

1. Formula for $2 \times 2$

$\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \frac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$

2. Cofactor Method (Adjugate)

$A^{-1} = \frac{1}{\det(A)}(\text{adj }A)$

where $\text{adj}(A) = C^T$ is the transpose of the cofactor matrix.

3. Row Reduction (Gaussian Elimination)

Augment $[A|I]$ and perform row operations:

$[A|I] \to [I|A^{-1}]$

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Example: $2 \times 2$ Inverse

$A = \begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix}$

$\det(A) = 6 - 5 = 1$

$A^{-1} = \frac{1}{1}\begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix} = \begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix}$

Verify: $AA^{-1} = \begin{pmatrix} 6-5 & -2+2 \\ 15-15 & -5+6 \end{pmatrix} = I$

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Properties

Property	Formula
Uniqueness	Inverse is unique (if exists)
Product	$(AB)^{-1} = B^{-1}A^{-1}$ (reversed order)
Transpose	$(A^T)^{-1} = (A^{-1})^T$
Double inverse	$(A^{-1})^{-1} = A$
Determinant	$\det(A^{-1}) = 1/\det(A)$

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Kernel (Null Space)

The kernel of $A$ is:

$\ker(A) = \{\mathbf{x} : A\mathbf{x} = \mathbf{0}\}$

Key results:

• $\ker(A) = \{\mathbf{0}\}$ $\Leftrightarrow$ $\det(A) \neq 0$ $\Leftrightarrow$ $A^{-1}$ exists
• Non-trivial kernel $\Leftrightarrow$ columns are linearly dependent

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Solving $A\mathbf{x} = \mathbf{y}$

If $\det(A) \neq 0$:

$\mathbf{x} = A^{-1}\mathbf{y}$

Row reduction method: Augment $[A|\mathbf{y}]$ and solve.

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When Inverse Doesn’t Exist

$\det(A) = 0$ means:

• Columns are linearly dependent
• $\ker(A) \neq \{\mathbf{0}\}$
• Either no solution or infinitely many solutions to $A\mathbf{x} = \mathbf{y}$

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Left and Right Inverses

For $m \times n$ matrices:

• Left inverse: $BA = I_n$ (if columns of $A$ are independent)
• Right inverse: $AB = I_m$ (if rows of $A$ are independent)

For square matrices: left inverse = right inverse = unique inverse.

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Why Matrix Inverse Matters

• Solve systems $A\mathbf{x} = \mathbf{b}$ directly: $\mathbf{x} = A^{-1}\mathbf{b}$
• Undo transformations (inverse rotation, inverse scaling)
• Essential for parameter estimation in statistics
• Foundation for understanding linear maps

Note: In practice, row reduction or decomposition methods are preferred for numerical stability. Direct inversion is rarely computationally optimal.