Linear Algebra/Inverses - Wikibooks, open books for an open world (2023)

Example 4.1

Where is the projection map

and is the embedding

the composition is the identity map on .

We say is a left inverse map of or, what is the same thing, that is a right inverse mapof .However, composition in the other order doesn't give the identity map— here is a vector that is not sent to itself under .

In fact, the projection has no left inverse at all.For, if were to be a left inverse of then we would have

for all of the infinitely many 's. But no function can send a single argument to more than one value.

Definition 4.2

A matrix is a left inverse matrix of the matrix if is the identity matrix. It is a right inverse matrix if is the identity. A matrix with a two-sided inverse is an invertible matrix. That two-sided inverse is called the inverse matrix and is denoted .

Lemma 4.3

If a matrix has both a left inverse and a right inverse then the two are equal.

Theorem 4.4

A matrix is invertible if and only if it is nonsingular.

Proof

(For both results.) Given a matrix , fix spaces of appropriate dimension for the domain and codomain. Fix bases for these spaces. With respect to these bases, represents a map . The statements are true about the map and therefore they are true about the matrix.

Lemma 4.5

A product of invertible matrices is invertible— if and are invertible and if is defined then is invertible and .

Proof

(This is just like the prior proof except that it requires two maps.)Fix appropriate spaces and bases and consider the represented maps and.Note that is a two-sided map inverse of sinceand.This equality is reflected in the matrices representing the maps, as required.

Example 4.6

We can find a left inverse for the matrix just given

by using Gauss' method to solve the resulting linear system.

Answer: , , , and .This matrix is actually the two-sided inverse of , as can easily be checked.With it we can solve the system () above byapplying the inverse.

Remark 4.7

Why solve systems this way, whenGauss' method takes less arithmetic(this assertion can be made precise by counting the number of arithmetic operations,as computer algorithm designers do)? Beyond its conceptual appeal of fitting into our program ofdiscovering how to represent the various map operations,solving linear systems by using the matrix inverse has at least two advantages.

First, once the work of finding an inverse has been done, solving a system with thesame coefficients but different constants is easy and fast: ifwe change the entries on the right of the system () then we get a related problem

with a related solution method.

In applications, solving many systems having the same matrix ofcoefficients is common.

Another advantage of inverses is that we can explore a system's sensitivity to changes in the constants.For example, tweaking the on the right of the system () to

can be solved with the inverse.

to show that changes by of the tweak while moves by of that tweak. This sort of analysis is used, for example, to decide how accurately data must be specified in a linear model to ensure that the solution has a desired accuracy.

Lemma 4.8

A matrix is invertible if and only if it can be written as the product of elementary reduction matrices. The inverse can be computed by applying to the identity matrix the same row steps, in the same order, as are used to Gauss-Jordan reduce the invertible matrix.

Proof

A matrix is invertible if and only if it is nonsingular and thus Gauss-Jordan reduces to the identity.By Corollary 3.22 this reduction canbe done with elementary matrices .This equation gives the two halves of the result.

First, elementary matrices are invertible and their inverses are alsoelementary.Applying to the left of both sides of that equation, then, etc., gives as the product ofelementary matrices (the is here to cover the trivial case).

Second, matrix inverses are unique and so comparison of the above equation with shows that . Therefore, applying to the identity, followed by , etc., yields the inverse of .

Example 4.9

To find the inverse of

we do Gauss-Jordan reduction, meanwhile performing the same operations onthe identity.For clerical convenience we write the matrix and the identity side-by-side,and do the reduction steps together.

This calculation has found the inverse.

Example 4.10

This one happens to start with a row swap.

Example 4.11

A non-invertible matrix is detected by the fact that the left half won'treduce to the identity.

Corollary 4.12

The inverse for a matrix exists and equals

if and only if .

Proof

This computation is Problem 10.

Problem 1

Supply the intermediate steps in Example 4.10.

Problem 2

Use Corollary 4.12 to decide if each matrix has an inverse.

Problem 3

For each invertible matrix in the prior problem, useCorollary 4.12 to find its inverse.

Problem 4

Find the inverse, if it exists, by using the Gauss-Jordan method.Check the answers for the matrices with Corollary 4.12.

Problem 5

What matrix has this one for its inverse?

Problem 6

How does the inverse operation interact with scalar multiplication and addition of matrices?

1. What is the inverse of ?
2. Is ?

Problem 7

Is ?

Problem 8

Is invertible?

Problem 9

For each real number let be represented with respect to thestandard bases by this matrix.

Show that .Show also that .

Problem 10

Do the calculations for the proof of Corollary 4.12.

Problem 11

Show that this matrix

has infinitely many right inverses.Show also that it has no left inverse.

Problem 12

In Example 4.1,how many left inverses has ?

Problem 13

If a matrix has infinitely many right-inverses, can it have infinitelymany left-inverses?Must it have?

Problem 14

Assume that is invertible and that is the zero matrix.Show that is a zero matrix.

Problem 15

Prove that if is invertible thenthe inverse commutes with a matrix if and only if itself commutes with that matrix .

Problem 16

Show that if is square and if is the zero matrixthen .Generalize.

Problem 17

Let be diagonal.Describe , , ... , etc.Describe , , ... , etc.Define appropriately.

Problem 18

Prove that any matrix row-equivalent to an invertible matrix is alsoinvertible.

Problem 19

The first question below appeared asProblem 15 in the Matrix Multiplication subsection.

1. Show that the rank of the product of two matrices is less thanor equal to the minimum of the rank of each.
2. Show that if and are square then if and only if .

Problem 20

Show that the inverse of a permutation matrix is its transpose.

Problem 21

The first two parts of this question appeared as Problem 12. of the Matrix Multiplication subsection

1. Show that .
2. A square matrix is symmetric if each entry equals the entry (that is, if the matrix equals its transpose).Show that the matrices and are symmetric.
3. Show that the inverse of the transpose is the transpose of the inverse.
4. Show that the inverse of a symmetric matrix is symmetric.

Problem 22

The items starting this question appeared asProblem 17 of the Matrix Multiplication subsection.

1. Prove that the composition of the projections is the zero map despite thatneither is the zero map.
2. Prove that the composition of the derivativesis the zero map despite that neither map is the zero map.
3. Give matrix equations representing each of the prior twoitems.

When two things multiply to give zero despitethat neither is zero, each is said to be a zero divisor.Prove that no zero divisor is invertible.

Problem 23

In real number algebra, there are exactly two numbers, and , that are their own multiplicative inverse.Does have exactly two solutions for matrices?

Problem 24

Is the relation "is a two-sided inverse of" transitive?Reflexive?Symmetric?

Problem 25

Prove: if the sum of the elements in each row of a squarematrix is , then the sum of the elements in each row of theinverse matrix is .(Wilansky 1951)