Chapter 5: Inverses

Chapter 5 — Inverses, Elementary Matrices, and the Invertible Matrix Theorem

A square matrix is invertible when its action can be completely undone. Algebraically, $A^{-1}A=AA^{-1}=I$. Geometrically, an invertible matrix does not collapse the plane or space into a lower-dimensional object.

inverse matrixGauss--Jordan on $[A\mid I]$elementary matricesno information loss

5.1

Undoing a matrix

Definition

For an $n\times n$ matrix $A$, an inverse is a matrix $A^{-1}$ satisfying

$$A^{-1}A=AA^{-1}=I_n.$$

If it exists, $A$ is invertible. If not, $A$ is singular.

Why it solves systems

When $Ax=b$ and $A$ is invertible, multiply both sides by $A^{-1}$:

$$x=A^{-1}b.$$

So an invertible matrix gives a unique solution for every right-hand side $b$.

5.2

The $2\times2$ inverse formula

For $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$, the determinant $ad-bc$ measures whether the two columns collapse onto one line. If $ad-bc\ne0$, then $$A^{-1}=\frac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}.$$

Interactive: compute a $2\times2$ inverse

$a$

$b$

$c$

$d$

5.3

Gauss--Jordan inverse calculator

For larger square matrices, row reduce the augmented matrix $[A\mid I]$. If the left half becomes $I$, then the right half is $A^{-1}$.

Interactive: row reduce $[A\mid I]$

Enter a square matrix $A$

5.4

Solving $Ax=b$ using an inverse

Interactive: solve a system

Matrix $A$

Vector $b$

5.5

The Invertible Matrix Theorem

System view

$Ax=b$ has exactly one solution for every $b\in\mathbb R^n$.

Homogeneous view

$Ax=0$ has only the trivial solution $x=0$.

RREF view

$\operatorname{rref}(A)=I_n$; every row and every column has a pivot.

Column view

The columns are independent and span $\mathbb R^n$.

Determinant view

$\det(A)\ne0$.

Geometry view

No dimension is collapsed: no information is lost.

5.6

Geometry of invertibility in $\mathbb R^2$

Move the entries of the matrix and watch how the grid transforms. If the image collapses to a line, the matrix is singular.

Interactive: transform the grid by $A$

$a_{11}$

$a_{12}$

$a_{21}$

$a_{22}$

5.7

Python connection

import numpy as np
A = np.array([[1, 1, 1], [1, 2, 3], [1, 3, 6]], dtype=float)
b = np.array([1, 2, 3], dtype=float)

A_inv = np.linalg.inv(A)      # compute inverse
x = np.linalg.solve(A, b)     # preferred for solving Ax=b
cond = np.linalg.cond(A)      # numerical sensitivity

Numerical warning: in applications, use solve(A,b) rather than explicitly computing $A^{-1}b$.

5.8

Quick checks

If $A$ is invertible and $Ax=0$, what is $x$?

$x=0$ only.

Can a nonsquare matrix have an inverse matrix in this chapter's sense?

No. The inverse matrix $A^{-1}$ is defined for square matrices only.

Why does determinant zero mean no inverse for a $2\times2$ matrix?

The inverse formula divides by $ad-bc$. If this number is zero, the columns collapse onto one line and the action cannot be undone.