Matrix Inverses

Chapter 2 • Section 2-4

"Looking for a way back? You need an inverse! 🔙"

↩ī¸ What is an Inverse?

The Definition

An $n \times n$ matrix $A$ is invertible if there exists a matrix $B$ such that:

$$ AB = I_n \quad \text{and} \quad BA = I_n $$
unique!

Theorem: Uniqueness of Inverse

If $A$ has an inverse, it is unique.
Proof: Suppose $B$ and $C$ are both inverses of $A$. Then $AB = I$ and $CA = I$.
$B = IB = (CA)B = C(AB) = CI = C$. Thus $B = C$. $\blacksquare$

Warning

Not all square matrices have inverses. A matrix without an inverse is called singular or non-invertible.

Example: The zero matrix is singular.

The $2 \times 2$ Shortcut

For a $2 \times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$:

The Formula

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

The quantity $ad-bc$ is called the determinant ($\det A$).

If $\det A = 0$, the matrix is not invertible!

Visual Mnemonic

a
b
c
d

1. Swap Blue (Diagonal)

2. Negate Red (Off-diagonal)

3. Divide by Determinant

🧮 Quick Inverse Win $10

Find the inverse of $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$. Note: $\det A = 4-6 = -2$.

The General Algorithm

For larger matrices ($3 \times 3$ and up), we use row reduction.

1

Augment the matrix with the Identity: $[A \mid I]$.

2

Row reduce until the left side becomes the Identity: $[I \mid A^{-1}]$.

3

If you can't get $I$ on the left, then $A$ is not invertible.

Concept Check

Problem

True or False: If $A^3 = 4I$, then $A$ is invertible.

True!

We need to find a matrix $B$ such that $AB = I$ and $BA = I$.

Given $A^3 = 4I$, we can rearrange it:

$A( \frac{1}{4} A^2 ) = I$ and $(\frac{1}{4} A^2) A = I$

Thus, $A$ is invertible and $A^{-1} = \frac{1}{4}A^2$.

Key Properties

Inverse of a Product

$$ (AB)^{-1} = B^{-1} A^{-1} $$
then shoes ($B$). To undo, you take off shoes ($B^{-1}$) then socks ($A^{-1}$).

Inverse of Transpose

$$ (A^T)^{-1} = (A^{-1})^T $$

You can swap the order of inverse and transpose.

Proof: We want to show $(A^{-1})^T$ is the inverse of $A^T$.
$A^T (A^{-1})^T = (A^{-1} A)^T = I^T = I$.
$(A^{-1})^T A^T = (A A^{-1})^T = I^T = I$. $\blacksquare$

🧠 Logic Check Win $20

If $A$ is invertible, what is $(A^{-1})^{-1}$?