Inverse Matrices

Overview

The inverse of a matrix $A$, denoted $A^{-1}$, is the matrix that when multiplied by $A$ gives the identity matrix. Not all matrices have inverses.

Definition

For a square matrix $A$, the inverse $A^{-1}$ satisfies:

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

Invertibility

A matrix $A$ is invertible (or non-singular) if $A^{-1}$ exists.

Conditions for Invertibility

A square matrix $A$ is invertible if and only if:

• $\det(A) \neq 0$
• $A$ has $n$ pivot positions (full rank)
• $A\mathbf{x} = \mathbf{0}$ has only the trivial solution
• Columns of $A$ are linearly independent
• Rows of $A$ are linearly independent

Finding the Inverse

2×2 Matrix Formula

For $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$:

$A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$

where $\det(A) = ad - bc$

Example

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

$\det(A) = 3(1) - 1(2) = 1$

$A^{-1} = \begin{bmatrix} 1 & -1 \\ -2 & 3 \end{bmatrix}$

Row Reduction Method

For larger matrices, augment $A$ with $I$ and row reduce:

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

Example

$\begin{bmatrix} 1 & 2 & \vert & 1 & 0 \\ 3 & 4 & \vert & 0 & 1 \end{bmatrix} \to \begin{bmatrix} 1 & 2 & \vert & 1 & 0 \\ 0 & -2 & \vert & -3 & 1 \end{bmatrix} \to \begin{bmatrix} 1 & 0 & \vert & -2 & 1 \\ 0 & 1 & \vert & 3/2 & -1/2 \end{bmatrix}$

$A^{-1} = \begin{bmatrix} -2 & 1 \\ 3/2 & -1/2 \end{bmatrix}$

Properties of Inverse

Property	Formula
Uniqueness	Inverse is unique if it exists
Involution	$(A^{-1})^{-1} = A$
Product	$(AB)^{-1} = B^{-1}A^{-1}$
Transpose	$(A^T)^{-1} = (A^{-1})^T$
Scalar	$(cA)^{-1} = (1/c)A^{-1}$
Power	$(A^n)^{-1} = (A^{-1})^n$

Product Inverse

Important: The inverse of a product reverses the order:

$(ABC)^{-1} = C^{-1}B^{-1}A^{-1}$

Singular Matrices

A singular matrix has no inverse.

Characteristics

• $\det(A) = 0$
• $A\mathbf{x} = \mathbf{0}$ has non-trivial solutions
• Columns are linearly dependent
• Not full rank

Example

$A = \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}$

$\det(A) = 1(4) - 2(2) = 0$

$A$ is singular (no inverse exists)

Solving Systems with Inverse

For the system $A\mathbf{x} = \mathbf{b}$ where $A$ is invertible:

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

Example

Solve:

$3x + y = 5$ $2x + y = 4$

$A = \begin{bmatrix} 3 & 1 \\ 2 & 1 \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 5 \\ 4 \end{bmatrix}$

$A^{-1} = \begin{bmatrix} 1 & -1 \\ -2 & 3 \end{bmatrix}$

$\mathbf{x} = A^{-1}\mathbf{b} = \begin{bmatrix} 1 & -1 \\ -2 & 3 \end{bmatrix}\begin{bmatrix} 5 \\ 4 \end{bmatrix} = \begin{bmatrix} 5 - 4 \\ -10 + 12 \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$

Solution: $x = 1$, $y = 2$

Inverse of Special Matrices

Diagonal Matrix

$D = \begin{bmatrix} d_1 & 0 & 0 \\ 0 & d_2 & 0 \\ 0 & 0 & d_3 \end{bmatrix} \quad D^{-1} = \begin{bmatrix} 1/d_1 & 0 & 0 \\ 0 & 1/d_2 & 0 \\ 0 & 0 & 1/d_3 \end{bmatrix}$

Orthogonal Matrix

$Q^{-1} = Q^T$

Upper Triangular Matrix

Inverse is also upper triangular (use back substitution).