- (a)
- (b)
- (c)
Additional Exercises for Ch 4: Matrices
Review Exercises
For the following pairs of matrices, determine if the sum is defined. If so, find the
sum.
Using only the properties given in Theorem th:propertiesofaddition and Theorem th:propertiesscalarmult, show that the additive
inverse of , , is unique.
Click on the arrow to see answer.
Suppose is also an additive inverse of . Then
Using only the properties given in Theorem th:propertiesofaddition and Theorem th:propertiesscalarmult, show that the zero
matrix, , is unique.
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Suppose is also an additive identity. Then
Using only the properties given in Theorem th:propertiesofaddition and Theorem th:propertiesscalarmult, show that
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Now add to both sides. Then .
Using only the properties given in Theorem th:propertiesofaddition and Theorem th:propertiesscalarmult, as well as previous
problems, show
Click on the arrow to see answer.
Therefore, from the uniqueness of the additive inverse proved in the above Problem addinvrstunique, it follows that .
Consider the matrices .
Find the following if possible. If it is not possible explain why.
- (a)
-
Click on the arrow to see answer.
- (b)
-
Click on the arrow to see answer.
- (c)
-
Click on the arrow to see answer.
Not possible
- (d)
-
Click on the arrow to see answer.
- (e)
-
Click on the arrow to see answer.
Not possible
- (f)
-
Click on the arrow to see answer.
Not possible
Consider the matrices
Find the following if possible. If it is not possible explain why.
- (a)
-
Click on the arrow to see answer.
- (b)
-
Click on the arrow to see answer.
Not possible
- (c)
-
Click on the arrow to see answer.
- (d)
-
Click on the arrow to see answer.
Not possible
- (e)
-
Click on the arrow to see answer.
- (f)
-
Click on the arrow to see answer.
Not possible
- (g)
-
Click on the arrow to see answer.
Not possible
- (h)
-
Click on the arrow to see answer.
Let , and Find the following if possible.
- (a)
-
Click on the arrow to see answer.
- (b)
-
Click on the arrow to see answer.
- (c)
-
Click on the arrow to see answer.
Not possible
- (d)
-
Click on the arrow to see answer.
- (e)
-
Click on the arrow to see answer.
Not possible
- (f)
-
Click on the arrow to see answer.
Let . Find all matrices, such that
Solution is: so the matrices are of the form
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Let Is it possible to choose such that If so, what should equal?
Thus you must have .
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Let Is it possible to choose such that If so, what should equal?
However, and so there is no possible choice of which will make these matrices
commute.
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Compute the following using block multiplication (all blocks are ).
Click the arrow to see the answer.
Click the arrow to see the answer.
Consider the matrices
Find the following if possible. If it is not possible explain why.
- (a)
- (b)
- (c)
- (d)
- (e)
- (f)
- (g)
Click on the arrow to see answer.
- (a)
- (b)
- (c)
- (d)
- (e)
- (f)
- (g)
- Not possible.
Show that the main diagonal of every skew symmetric matrix consists of only zeros.
Recall that the main diagonal consists of every entry of the matrix which is of the
form .
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If is skew-symmetric then It follows that and so each .
Let
Find if possible. If does not exist, explain why.
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does not exist. The reduced row echelon form of this matrix is
If possible, find the inverse of each matrix by using the row-reduction procedure. If
you find that does not have an inverse, leave the answer matrix blank.
Let
Find if possible. If does not exist, explain why.
Click on the arrow to see answer.
The reduced row echelon form is . There is no inverse.
Using the inverse of the matrix, find the solution to the systems:
- (a)
- (b)
Using the inverse of the matrix, find the solution to the systems:
- (a)
-
Click on the arrow to see answer.
- (b)
-
Prove that if exists and then .
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Multiply on both sides on the left by Thus
Show that if exists for an matrix, then it is unique. That is, if and then
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Let . Suppose a row operation is applied to and the result is . Find the elementary
matrix that represents this row operation.
Let . Suppose a row operation is applied to and the result is . Find the elementary
matrix that represents this row operation.
Let . Suppose a row operation is applied to and the result is . Find the elementary
matrix that represents this row operation.
Let . Suppose a row operation is applied to and the result is .
- (a)
- Find the elementary matrix such that .
- (b)
- Find the inverse of , , such that .
Let . Suppose a row operation is applied to and the result is .
- (a)
- Find the elementary matrix such that .
- (b)
- Find the inverse of , , such that .
Let . Suppose a row operation is applied to and the result is .
- (a)
- Find the elementary matrix such that .
- (b)
- Find the inverse of , , such that .
Let . Suppose a row operation is applied to and the result is .
- (a)
- Find the elementary matrix such that .
- (b)
- Find the inverse of , , such that .
Find an factorization of the coefficient matrix and use it to solve the system of
equations.
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An factorization of the coefficient matrix is First solve which gives Then solve which says that and
Find an factorization of the coefficient matrix and use it to solve the system of
equations.
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An factorization of the coefficient matrix is First solve which yields . Next solve This yields
Find an factorization of the coefficient matrix and use it to solve the system of
equations.
Click on the arrow to see answer.
An factorization of the coefficient matrix is First solve Solution is: Next solve Solution is: .
Is there only one factorization for a given matrix?
Click on the arrow to see answer.
Consider the equation
Look for all possible factorizations.Click the arrow for the answer.
Sometimes there is more than one factorization as is the case in this example. The given equation clearly gives an factorization. However, it appears that the following equation gives another factorization.
Challenge Exercises
Show that, if a (possibly nonhomogeneous) system of equations is consistent
and has more variables than equations, then it must have infinitely many
solutions.
Assume that a system of linear equations has at least two distinct solutions and
.
- a.
- Show that is a solution for every .
- b.
- Show that implies . See Example exa:002159.
- c.
- Deduce that has infinitely many solutions.
Click on the arrow to see answer.
- (a)
- If , then . So . But is not zero (because and are distinct), so by Example exa:002159.
- (a)
- Let be a matrix with all entries on and below the main diagonal zero. Show that .
- (b)
- Generalize to the case and prove your answer.
Let denote the matrix with -entry equal to and all other entries . Show
that:
- (a)
- .
- (b)
- .
- (c)
- If is , then .
- (d)
- If , then for all , , , and .
Click on the arrow to see answer.
Using parts (c) and (b) gives . The only nonzero term occurs when and , so .
A matrix of the form , where is a number, is called an scalar matrix.
- (a)
- Show that each scalar matrix commutes with every matrix.
- (b)
- Show that is a scalar matrix if it commutes with every matrix. See part (d.) of Exercise ex:ex2_suppl_6.
Click on the arrow to see answer.
- (a)
- If , then by 6(d). But then if , so if . If , then is independent of . Thus for all .
If is , show that if and only if for some or
for some .
Click on the arrow to see answer.
If , then , for some . Use
Let and be elementary matrices obtained from the identity matrix by adding
multiples of row to rows and . If and , show that .
Show that the following are equivalent for matrices , :
- (a)
- , , and are all invertible and
- (b)
- is invertible and where .
Octave Exercises
Use Octave to check your work on Problems prb:4.9 to prb:4.11. The first steps of prb:4.9 are in the code
below. See if you can finish the rest of the problem.
To use Octave, go to the Sage Math Cell Webpage, copy the code below into the cell, select OCTAVE as the language, and press EVALUATE.
% Matrix multiplication
A=[1 2 3; 2 1 7];
B=[3 -1 2; -3 2 1];
C=[1 2; 3 1];
-3*A
3*B-A
A*C %This is not possible.
%After the error, remove this and try C*B
Use Octave to check your work on Problems prb:4.24 to prb:4.27. The first steps of prb:4.27 are in the code
below. See if you can finish the rest of the problem.
To use Octave, go to the Sage Math Cell Webpage, copy the code below into the cell, select OCTAVE as the language, and press EVALUATE.
% Matrix arithmetic
A=[1 2 ; 3 2; 1 -1];
B=[2 -5 2; -3 2 1];
C=[1 2; 5 0];
D=[-1 1; 4 -3];
E=[1;3];
-3*transpose(A)
3*B-transpose(A)
transpose(E)*B
E*transpose(E)
Use Octave to check your work on Problems prb:4.35 to prb:4.38 and Problems prb:4.40 to prb:4.44. Problem prb:4.36 is in
the code below.
To use Octave, go to the Sage Math Cell Webpage, copy the code below into the cell, select OCTAVE as the language, and press EVALUATE.
% Finding an inverse
A=[0 1; 5 3];
% We can find an inverse by augmenting with the identity matrix and performing Gauss-Jordan elimination.
M = [A eye(length(A))]
rref(M)
% We can also use the Octave command to compute the inverse.
inv(A)
% and, to check our work...
ans*A
Bibliography
The Review Exercises come from the end of Chapter 2 of Ken Kuttler’s A First Course in Linear Algebra. (CC-BY)
Ken Kuttler, A First Course in Linear Algebra, Lyryx 2017, Open Edition, pp. 90–98, 104–106.
The Challenge Exercises come from the end of Chapter 2 of Keith Nicholson’s Linear Algebra with Applications. (CC-BY-NC-SA)
W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2018, Open Edition, pp. 143–144.