Matrix Inverses

Inverse

Definition

Let . If there exists a such that , then WLOG, is invertible and is the inverse of .

The inverse of is denoted by

(see matrix multiplication, identity matrix)

To see what a matrix inverse represents, see Inverse Linear Transformations.

Wrong

Do not write . This is wrong

Example

Given

Then

Inverse Existence

Theorem

Let . if and only if . Moreover, , and and

(See rank, logical and, determinant)

Proof

Let be such that . Let be such that (see matrix-vector product)

Show
Since ,

(see zero vector)

so is then only solution to the homogeneous system . Thus, by [Math/Linear Algebra/Matrices/Rank#System-Rank Theorem|system-rank theorem].

Show
Since and has rows, the [Math/Linear Algebra/Matrices/Rank#System-Rank Theorem|system-rank theorem] guarantees that we will find such that . Then

so for all . Thus, by the matrices equal theorem.

Finally, since , it follows that by the first part of the proof, with and interchanged.

Intuition

A matrix must be full rank, otherwise it squishes some vectors into a lower dimension. You cannot "unsquish" these vectors, since the linear transformation would no longer be a function (each input would have multiple outputs).

Inverse Uniqueness

Theorem

Let be invertible. If are both inverses of , then

See logical and

Proof

We prove the uniqueness of an inverse
Assume that both and $C are inverses of . Then and . We have

Hence, if is invertible, the inverse of is unique.

Matrix Inversion Algorithm

Consider . If is invertible, then there exists an such that

Thus

We have three systems of equations with the same coefficient matrix, so we construct an augmented matrix

(Since is the identity matrix)

We must consider 2 cases:

If the RREF of is , thenwhere is the result of Gauss-Jordan elimination on .
From this, we see thatHenceSo
Else
, and is not invertible, since if were invertible, it would have by the "inverse existence" theorem

Summary

For , to see if is invertible, and to compute , carry the matrix to RREF.

If the RREF of is for some :
Else: is not invertible

Check Your Work

If an inverse is found, you can check to ensure that

Inverse exists

Given

Find if it exists

Notice how we started with the identity matrix as the coefficient matrix. and ended with the identity matrix as the constant matrix.

We conclude that is invertible and

Inverse does not exist

Given

Find if it exists

We see that the RREF of is NOT the identity matrix, so is not invertible (see that )

Properties of Matrix Inverses

Properties

Let be invertible and let with . Then

(see transpose)
(think about it)

Properties - Cancellation

Let be invertible

Left cancellation:
Right cancellation:

Invertible Matrix Theorem

Let . The following are equivalent:

is invertible
The RREF of is
For all , the system is consistent, and has a unique solution
is invertible (see transpose)
(see null space)
The columns of form a linearly independent set
The columns of span
(see column space)
The rows of form a linearly independent set
The rows of span
(see row space)