Linear Algebra/Inverses/Solutions

SolutionsEdit

Problem 1

Supply the intermediate steps in Example 4.10.

Answer

Here is one way to proceed.

 
 
This exercise is recommended for all readers.
Problem 2

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

  1.  
  2.  
  3.  
Answer
  1. Yes, it has an inverse:  .
  2. Yes.
  3. No.
This exercise is recommended for all readers.
Problem 3

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

Answer
  1.  
  2.  
  3. The prior question shows that no inverse exists.
This exercise is recommended for all readers.
Problem 4

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

  1.  
  2.  
  3.  
  4.  
  5.  
  6.  
Answer
  1. The reduction is routine.
     
    This answer agrees with the answer from the check.
     
  2. This reduction is easy.  
     
    The check agrees.
     
  3. Trying the Gauss-Jordan reduction
     
    shows that the left side won't reduce to the identity, so no inverse exists. The check   agrees.
  4. This produces an inverse.
     
     
  5. This is one way to do the reduction.
     
     
  6. There is no inverse.
     
    As a check, note that the third column of the starting matrix is   times the second, and so it is indeed singular and therefore has no inverse.
This exercise is recommended for all readers.
Problem 5

What matrix has this one for its inverse?

 
Answer

We can use Corollary 4.12.

 
Problem 6

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

  1. What is the inverse of  ?
  2. Is  ?
Answer
  1. The proof that the inverse is   (provided, of course, that the matrix is invertible) is easy.
  2. No. For one thing, the fact that   has an inverse doesn't imply that   has an inverse or that   has an inverse. Neither of these matrices is invertible but their sum is.
     
    Another point is that just because   and   each has an inverse doesn't mean   has an inverse; here is an example.
     
    Still a third point is that, even if the two matrices have inverses, and the sum has an inverse, doesn't imply that the equation holds:
     
    but
     
    and   does not equal  .
This exercise is recommended for all readers.
Problem 7

Is  ?

Answer

Yes:  .

Problem 8

Is   invertible?

Answer

Yes, the inverse of   is  .

Problem 9

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

 

Show that  . Show also that  .

Answer

One way to check that the first is true is with the angle sum formulas from trigonometry.

 

Checking the second equation in this way is similar.

Of course, the equations can be not just checked but also understood by recalling that   is the map that rotates vectors about the origin through an angle of   radians.

Problem 10

Do the calculations for the proof of Corollary 4.12.

Answer

There are two cases. For the first case we assume that   is nonzero. Then

 

shows that the matrix is invertible (in this   case) if and only if  . To find the inverse, we finish with the Jordan half of the reduction.

 

The other case is the   case. We swap to get   into the   position.

 

This matrix is nonsingular if and only if both   and   are nonzero (which, under the case assumption that  , holds if and only if  ). To find the inverse we do the Jordan half.


 

(Note that this is what is required, since   gives that  ).

Problem 11

Show that this matrix

 

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

Answer

With   a   matrix, in looking for a matrix   such that the combination   acts as the   identity we need   to be  . Setting up the equation

 

and solving the resulting linear system

 

gives infinitely many solutions.

 

Thus   has infinitely many right inverses.

As for left inverses, the equation

 

gives rise to a linear system with nine equations and four unknowns.

 


This system is inconsistent (the first equation conflicts with the third, as do the seventh and ninth) and so there is no left inverse.

Problem 12

In Example 4.1, how many left inverses has  ?

Answer

With respect to the standard bases we have

 

and setting up the equation to find the matrix inverse

 

gives rise to a linear system.

 


There are infinitely many solutions in   to this system because two of these variables are entirely unrestricted

 

and so there are infinitely many solutions to the matrix equation.

 

With the bases still fixed at  , for instance taking   and   gives a matrix representing this map.

 

The check that   is the identity map on   is easy.

Problem 13

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

Answer

By Lemma 4.3 it cannot have infinitely many left inverses, because a matrix with both left and right inverses has only one of each (and that one of each is one of both— the left and right inverse matrices are equal).

This exercise is recommended for all readers.
Problem 14

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

Answer

The associativity of matrix multiplication gives on the one hand  , and on the other that  .

Problem 15

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

Answer

Multiply both sides of the first equation by  .

This exercise is recommended for all readers.
Problem 16

Show that if   is square and if   is the zero matrix then  . Generalize.

Answer

Checking that when   is multiplied on both sides by that expression (assuming that   is the zero matrix) then the result is the identity matrix is easy. The obvious generalization is that if   is the zero matrix then  ; the check again is easy.

This exercise is recommended for all readers.
Problem 17

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

Answer

The powers of the matrix are formed by taking the powers of the diagonal entries. That is,   is all zeros except for diagonal entries of  ,  , etc. This suggests defining   to be the identity matrix.

Problem 18

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

Answer

Assume that   is row equivalent to   and that   is invertible. Because they are row-equivalent, there is a sequence of row steps to reduce one to the other. That reduction can be done with matrices, for instance,   can be changed by row operations to   as  . This equation gives   as a product of invertible matrices and by Lemma 4.5 then,   is also invertible.

Problem 19

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

  1. Show that the rank of the product of two matrices is less than or equal to the minimum of the rank of each.
  2. Show that if   and   are square then   if and only if  .
Answer
  1. See the answer to Problem 15 in the Matrix Multiplication subsection.
  2. We will show that both conditions are equivalent to the condition that the two matrices be nonsingular. As   and   are square and their product is defined, they are equal-sized, say  . Consider the   half. By the prior item the rank of   is less than or equal to the minimum of the rank of   and the rank of  . But the rank of   is  , so the rank of   and the rank of   must each be  . Hence each is nonsingular. The same argument shows that   implies that each is nonsingular.
Problem 20

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

Answer

Inverses are unique, so we need only show that it works. The check appears above as Problem 9 of the Mechanics of Matrix Multiplication subsection.

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.
Answer
  1. See the answer for Problem 12 of the Matrix Multiplication subsection.
  2. See the answer for Problem 12 of the Matrix Multiplication subsection.
  3. Apply the first part to   to get  .
  4. Apply the prior item with  , as   is symmetric.
This exercise is recommended for all readers.
Problem 22

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

  1. Prove that the composition of the projections   is the zero map despite that neither is the zero map.
  2. Prove that the composition of the derivatives   is the zero map despite that neither map is the zero map.
  3. Give matrix equations representing each of the prior two items.

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

Answer

For the answer to the items making up the first half, see Problem 17 of the Matrix Multiplication subsection. For the proof in the second half, assume that   is a zero divisor so there is a nonzero matrix   with   (or else  ; this case is similar), If   is invertible then   but also  , contradicting that   is nonzero.

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?

Answer

No, there are at least four.

 
Problem 24

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

Answer

It is not reflexive since, for instance,

 

is not a two-sided inverse of itself. The same example shows that it is not transitive. That matrix has this two-sided inverse

 

and while   is a two-sided inverse of   and   is a two-sided inverse of  , we know that   is not a two-sided inverse of  . However, the relation is symmetric: if   is a two-sided inverse of   then   and therefore   is also a two-sided inverse of  .

Problem 25

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

Answer

This is how the answer was given in the cited source.

Let   be  , non-singular, with the stated property. Let   be its inverse. Then for  ,

 

(  is singular if  ).

ReferencesEdit

  • Wilansky, Albert (Nov. year=1951), "The Row-Sum of the Inverse Matrix", American Mathematical Monthly (American Mathematical Society) 58 (9): 614 .