Linear Algebra/Inverses/Solutions
Solutions
edit- This exercise is recommended for all readers.
- Problem 2
Use Corollary 4.12 to decide if each matrix has an inverse.
- Answer
- Yes, it has an inverse: .
- Yes.
- 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
- 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.
- Answer
- The reduction is routine.
- This reduction is easy.
- Trying the Gauss-Jordan reduction
- This produces an inverse.
-
- This is one way to do the reduction.
-
- There is no inverse.
- This exercise is recommended for all readers.
- Problem 6
How does the inverse operation interact with scalar multiplication and addition of matrices?
- What is the inverse of ?
- Is ?
- Answer
- The proof that the inverse is (provided, of course, that the matrix is invertible) is easy.
- 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.
- 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.
- Show that the rank of the product of two matrices is less than or equal to the minimum of the rank of each.
- Show that if and are square then if and only if .
- Answer
- See the answer to Problem 15 in the Matrix Multiplication subsection.
- 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
- Show that .
- 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.
- Show that the inverse of the transpose is the transpose of the inverse.
- Show that the inverse of a symmetric matrix is symmetric.
- Answer
- See the answer for Problem 12 of the Matrix Multiplication subsection.
- See the answer for Problem 12 of the Matrix Multiplication subsection.
- Apply the first part to to get .
- 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.
- Prove that the composition of the projections is the zero map despite that neither is the zero map.
- Prove that the composition of the derivatives is the zero map despite that neither map is the zero map.
- 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 ).