Chapter 11. Invertible Matrices

An invertible matrix is a square matrix that has a matrix inverse. The inverse reverses the action of the original matrix. If $A$ sends $x$ to $Ax$, then $A^{-1}$ sends $Ax$ back to $x$. Formally, an $n\times n$ matrix $A$ is invertible if there exists an $n\times n$ matrix $B$ such that $AB=BA=I_n$. In that case, $B$ is unique and is written $A^{-1}$.

11.1 Definition

Let $A$ be an $n\times n$ matrix over a field $F$. The matrix $A$ is invertible if there exists an $n\times n$ matrix $B$ such that

and

The matrix $B$ is called the inverse of $A$. It is denoted by

Thus

and

The identity matrix plays the role of $1$ in matrix multiplication. An invertible matrix is therefore a matrix that has a multiplicative inverse.

11.2 Why the Matrix Must Be Square

The usual inverse is defined for square matrices.

If $A$ is $m\times n$, then a product $AB$ can equal $I_m$ only if $B$ is $n\times m$. A product $BA$ can equal $I_n$ only if the same $B$ is $n\times m$. For both identities

and

to hold as a two-sided inverse in the ordinary finite-dimensional setting, the dimensions must match in the square case. Non-square matrices may have one-sided inverses in special cases, but they do not have ordinary two-sided inverses of the same kind used for square matrices.

11.3 The Meaning of an Inverse

If $A$ is invertible, then the transformation

can be undone.

Starting with a vector $x$, apply $A$:

Then apply $A^{-1}$:

By associativity,

Thus $A^{-1}$ recovers the original vector.

Likewise,

for every vector $y\in F^n$.

11.4 Uniqueness of the Inverse

If a matrix has an inverse, that inverse is unique.

Suppose $B$ and $C$ are both inverses of $A$. Then

and

We show that $B=C$:

Since $AC=I$,

By associativity,

Since $BA=I$,

Therefore the inverse is unique.

This justifies the notation $A^{-1}$.

11.5 Invertibility and Linear Systems

Invertibility gives the simplest solution rule for square systems.

If

and $A$ is invertible, then multiply both sides by $A^{-1}$:

Since

we get

Thus an invertible coefficient matrix gives a unique solution for every right-hand side $b$. Conversely, if every system $Ax=b$ has a unique solution, then $A$ is invertible. This equivalence is one form of the invertible matrix theorem.

11.6 Singular Matrices

A square matrix that is not invertible is called singular.

A singular matrix cannot be undone. The transformation $x\mapsto Ax$ loses information. In geometric terms, it collapses at least one nonzero direction. In algebraic terms, the homogeneous equation

has a nonzero solution.

For example,

is singular. Its second row is twice its first row, so the two rows do not give two independent constraints.

Indeed,

A nonzero vector has been sent to zero, so $A$ cannot be invertible.

11.7 The Invertible Matrix Theorem

For an $n\times n$ matrix $A$ over a field, the following statements are equivalent:

These conditions are either all true or all false for a given square matrix.

11.8 Invertibility and Row Reduction

A square matrix $A$ is invertible exactly when its reduced row echelon form is the identity matrix.

For example,

Row reduce:

using

Then

using

Then

using

Since $A$ row reduces to $I_2$, it is invertible.

11.9 Computing the Inverse by Row Reduction

To compute $A^{-1}$, form the augmented matrix

Then row reduce. If the left side becomes $I_n$, the right side becomes $A^{-1}$:

If the left side cannot be reduced to $I_n$, then $A$ is singular.

This method works because each row operation is equivalent to multiplying on the left by an elementary matrix. Reducing $A$ to $I_n$ records the inverse operation on the identity side.

11.10 Example: Computing an Inverse

Let

Form

Use

Then

Use

Then

Use

Then

Therefore

Check:

11.11 Formula for a $2\times 2$ Inverse

Let

Then $A$ is invertible exactly when

In that case,

The scalar $ad-bc$ is the determinant of $A$.

For example,

has determinant

Therefore

11.12 Invertibility and Determinants

For a square matrix over a field,

if and only if

If

then $A$ is singular.

The determinant criterion is often useful theoretically. In computation, however, solving systems by row reduction or decomposition is usually preferred to explicitly computing an inverse. Matrix inversion is often unnecessary when the goal is only to solve $Ax=b$.

11.13 Products of Invertible Matrices

If $A$ and $B$ are invertible $n\times n$ matrices, then $AB$ is invertible, and

The order reverses.

To verify this, compute:

Also,

Thus $B^{-1}A^{-1}$ is the inverse of $AB$.

11.14 Powers of an Invertible Matrix

If $A$ is invertible, then every positive power

is invertible, and

We also define negative powers by

Thus

This notation is meaningful only when $A$ is invertible.

11.15 Transposes of Invertible Matrices

If $A$ is invertible, then $A^T$ is invertible, and

To see this, transpose the identity

Using the rule

we get

Similarly, transposing

gives

Therefore $(A^{-1})^T$ is the inverse of $A^T$.

11.16 Elementary Matrices and Inverses

Every elementary row operation corresponds to multiplication by an elementary matrix.

Each elementary matrix is invertible because each elementary row operation is reversible.

For example, the row operation

on a $2\times 2$ matrix is represented by

The inverse operation is

represented by

Indeed,

11.17 Invertible Matrices as a Group

The set of all invertible $n\times n$ matrices over a field $F$ is denoted

It is called the general linear group.

This set is closed under matrix multiplication. It contains the identity matrix. Every element has an inverse. Matrix multiplication is associative. Thus $GL_n(F)$ forms a group under multiplication.

This group is important because it represents all invertible linear changes of coordinates in $F^n$.

11.18 Geometric Meaning

An invertible matrix represents a transformation that preserves dimension.

It may rotate, reflect, shear, scale, or combine these actions. But it cannot collapse a line to a point, a plane to a line, or a three-dimensional region to a plane.

For example,

is invertible. It stretches the $x$-direction by $2$ and the $y$-direction by $3$. Its inverse is

The inverse reverses the stretching.

By contrast,

is singular. It sends

to

It collapses the plane onto the $x$-axis, losing the $y$-coordinate. No inverse can recover information that has been lost.

11.19 Solving Without Explicit Inversion

Although the formula

is mathematically correct, it is often not the best computational method.

In numerical computation, one usually solves

by elimination or factorization, such as LU decomposition, rather than computing $A^{-1}$ explicitly. Computing an inverse can cost extra work and may introduce additional numerical error.

Thus the inverse is conceptually important, but direct inversion is not always computationally preferred.

11.20 Common Mistakes

11.21 Summary

An invertible matrix is a square matrix whose action can be reversed. Its inverse $A^{-1}$ satisfies

Invertibility has many equivalent forms:

Invertible matrices are the algebraic model of reversible linear transformations. They are central to solving systems, changing coordinates, understanding dimension, and studying linear maps.