Find the indicated entry of the matrix,
if it is defined.
Answer
Not defined.
This exercise is recommended for all readers.
Problem 2
Give the size of each matrix.
Answer
This exercise is recommended for all readers.
Problem 3
Do the indicated vector operation, if it is defined.
Answer
Not defined.
This exercise is recommended for all readers.
Problem 4
Solve each system using matrix notation.
Express the solution using vectors.
Answer
This reduction
leaves leading and free.
Making the parameter, we have so the solution
set is
This reduction
gives the unique solution , .
The solution set is
This use of Gauss' method
leaves and leading with free.
The solution set is
This reduction
shows that the solution set is a singleton set.
This reduction is easy
and ends with and leading, while and are
free.
Solving for gives and substitution shows
that so ,
making the solution set
The reduction
shows that there is no solution— the solution set is empty.
This exercise is recommended for all readers.
Problem 5
Solve each system using matrix notation.
Give each solution set in vector notation.
Answer
This reduction
ends with and leading while is free.
Solving for gives , and then substitution
shows that .
Hence the solution set is
This application of Gauss' method
leaves , , and leading.
The solution set is
This row reduction
ends with and free.
The solution set is
Gauss' method done in this way
ends with , , and free.
Solving for shows that and then
substitution
shows that
and so the solution set is
This exercise is recommended for all readers.
Problem 6
The vector is in the set.
What value of the parameters produces that vector?
,
,
,
Answer
For each problem we get a system of linear equations by looking at the
equations of components.
The second components show that , the third
components show that .
,
Problem 7
Decide if the vector is in the set.
,
,
,
,
Answer
For each problem we get a system of linear equations by looking at the
equations of components.
Yes; take .
No; the system with equations and
has no solution.
Yes; take .
No.
The second components give .
Then the third components give .
But the first components don't check.
Problem 8
Parametrize the solution set of this one-equation system.
Answer
This system has equation.
The leading variable is , the other variables are free.
This exercise is recommended for all readers.
Problem 9
Apply Gauss' method to the left-hand side to solve
for , , , and , in terms of the
constants , , and . Note that will be a free variable.
Use your answer from the prior part to solve this.
Answer
Gauss' method here gives
leaving free.
Solve: ,
and so
, and
Therefore the solution set is this.
Plug in with , , and .
This exercise is recommended for all readers.
Problem 10
Why is the comma needed in the notation ""
for matrix entries?
Answer
Leaving the comma out, say by writing ,
is ambiguous because it could mean or .
This exercise is recommended for all readers.
Problem 11
Give the matrix whose
-th entry is
;
to the power.
Answer
Problem 12
For any matrix , the
transpose
of , written
, is the matrix whose columns are the rows of .
Find the transpose of each of these.
Answer
This exercise is recommended for all readers.
Problem 13
Describe all functions
such that and .
Describe all functions
such that .
Answer
Plugging in and gives
so the set of functions is
.
Putting in gives
so the set of functions is
.
Problem 14
Show that any set of five points from the plane lie on a
common conic section, that is, they all satisfy some equation of the
form where some of
are nonzero.
Answer
On plugging in the five pairs we get a system with the
five equations and six unknowns , ..., .
Because there are more unknowns than equations, if no inconsistency
exists among the equations then there are infinitely many solutions
(at least one variable will end up free).
But no inconsistency can exist because , ..., is a
solution (we are only using this zero solution to show that the system
is consistent— the prior paragraph shows that
there are nonzero solutions).
Problem 15
Make up a four equations/four unknowns system having
a one-parameter solution set;
a two-parameter solution set;
a three-parameter solution set.
Answer
Here is one— the fourth equation is redundant
but still OK.
Here is one.
This is one.
? Problem 16
Solve the system of equations.
For what values of does the system fail to have solutions, and
for what values of are there infinitely many solutions?
This is how the answer was given in the cited source.
Formal solution of the system yields
If and , then the system has the single
solution
If , or if , then the formulas are meaningless; in the
first instance we arrive at the system
which is a contradictory system.
In the second instance we have
which has an infinite number of solutions (for example, for
arbitrary, ).
Solution of the system yields
Here, is , the system has the single solution
, .
For and , we obtain the systems
both of which have an infinite number of solutions.
? Problem 17
In air a gold-surfaced sphere weighs
grams.
It is known that it may contain one or more of the metals aluminum,
copper, silver, or lead.
When weighed successively under standard conditions in water, benzene,
alcohol, and glycerine its respective weights are , ,
, and grams.
How much, if any, of the forenamed metals does it contain if the
specific gravities of the designated substances are taken to be as follows?
This is how the answer was given in the cited source.
Let , , , , be the volumes in
of Al, Cu, Pb, Ag, and Au, respectively, contained in
the sphere, which we assume to be not hollow.
Since the loss of weight in water (specific gravity ) is
grams, the volume of the sphere is .
Then the data, some of which is superfluous, though consistent, leads to
only independent equations, one relating volumes and the
other, weights.
Clearly the sphere must contain some aluminum to bring its mean specific
gravity below the specific gravities of all the other metals.
There is no unique result to this part of the problem, for the amounts
of three metals may be chosen arbitrarily, provided that the choices
will not result in negative amounts of any metal.
If the ball contains only aluminum and gold, there are
of gold and of aluminum.
Another possibility is each of Cu, Au, Pb, and
Ag and of Al.