Hello! Once again I'm preparing for my Linear Algebra final and am
having trouble understanding these problems. If you know how to solve
some of them, please help me out.

1) Use the equation ax^2 + bx + c as a model to find a
determinant that is equal to ax^3 + bx^2 + cx + d. (Note: this problem is in
the section in my text called "Determinant of a Matrix".)

2) Prove the property.

|1+a    1     1|
| 1    1+b    1|   = abc(1 + 1/a + 1/b + 1/c). a, b, and c do not equal 0.
| 1     1   1+c|

3) If A is an idempotent matrix (A^2 = A), prove that the determinant
of A is either 0 or 1.

4) Please prove the follwing formula for a nonsingular n X n matrix A.
Assume n> or = to 3.

adj[adj(A)] = (|A|^(n-2))*(A). (In section "Adjoint of a Matrix".)

5) Verify the following system of linear equations in cosA, cosB, and
cosC for the triangle shown (note: side b is opposite angle B, side a
is opposite angle A, and side c is opposite angle C).

        ccosB + bcosC = a
ccosA         + acosC = b
bcosA + acosB         = c

Last one:

6) Determine whether the set, together with the indicated operations,
is a vector space. If it is not, identify at least one of the ten
vector space axioms that fails.

C[0,1], the set of all continuous functions defined on the interval
[0,1], with the standard operations.

Thank you very much for your time.

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Clarification of Question by alison28a-ga on 01 May 2005 07:36 PDT

Please try to answer this soon, as my final is tomorrow. I will add a
nice tip for a quicker response. Thanks again.

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Clarification of Question by alison28a-ga on 01 May 2005 21:13 PDT

livioflores or anyone out there: please answer this ASAP (preferably
before 2:00am). I will provide a generous tip.

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Clarification of Question by alison28a-ga on 01 May 2005 22:39 PDT

livioflores-- thanks for answering what you can. I really appreciate your help.

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Clarification of Question by alison28a-ga on 01 May 2005 22:55 PDT

On question 1, I should have given you more info.

57. verify the equation.

|x   0  c|
|-1  x  b| = ax^2 + bx + c
|0  -1  a|

Then QUESTION 1 says use the equation given in exercise 57 (instead of
what I wrote). Does that help at all?

Also, what do you mean when you refer to exercise 69? Thanks, and feel
free to respond for money.

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Clarification of Question by alison28a-ga on 01 May 2005 23:34 PDT

livioflores, last question (you need some sleep!):

In 6) is  the set a vector space? Or is there a property that isn't satisfied?

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Clarification of Question by alison28a-ga on 02 May 2005 01:25 PDT

Would this be right for 4)

adj[adj(A)] = adj[|A|^n |A^-1|]

=[|A|^n |A^-1|]^n * [|A|^n |A^-1|]^-1
=|A|^nsqaured * |A^-n| * |A^-n| * |A^1|
=|A|^nsquared * |A^-2n| * A

=  |A|^nsquared
   ---         * A
   |A^2n|

=  |A|^n-2 * A

Hi!!

If the question 1 can be rewritten as:
use the equation given in exercise 57 to find a determinant that is
equal to ax^3 + bx^2 + cx + d.

We have that:

      |x   0  c|
|A| = |-1  x  b| = ax^2 + bx + c
      |0  -1  a|

What I interpret is that we must find a matrix B such that:
|B| = ax^3 + bx^2 + cx + d

Recall some determinant properties:
If B is obtained by multiplying any row or column by k from A, then:
|B| = k*|A| 

This what I found:
Replace in A the value c by (c+d/x), we have a matrix D such that:

      |x   0  c+d/x |
|D| = |-1  x    b   | = ax^2 + bx + c + d/x
      |0  -1    a   |


Now if the matrix B is such that the last row is obtained by
multiplying it by x from D, we have that:

      |x   0  c+d/x |
|B| = |-1  x    b   | = x*|D| = 
      |0  -x    ax  |

    = x*(ax^2 + bx + c + d/x) =
    = ax^3 + bx^2 + cx + d


         ----------------------------------

Exercise 69 is a typo (number 9 instead of the parenthesis), so it
meas exercise 6).

         ----------------------------------

In 6) is  the set a vector space? Or is there a property that isn't satisfied?

In my opinion it is a vector space.

Properties of continuous functions:
If f and g are continuous, then f+g is continuous.
I f is continuous and k is a real number, then:
k*f is continous

See the following page for axioms:
"Vector Spaces"
http://www.cs.ut.ee/~toomas_l/linalg/lin1/node5.html

Now you can see easily that all the axioms are satisfied, then C[0,1]
is a vector space.

  ----------------------------------------------

Thank you for giving me the opportunity to answer this question. Feel
free to request for a clarification if you need it.

Best regards.
livioflores-ga

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Clarification of Answer by livioflores-ga on 02 May 2005 07:57 PDT

Thank you so much for the confidence, the good rating and your generous tips.
Good luck in your exams!!

I forgot add to the answer the solutions posted as a comment, so here they are:

2) Prove the property.

       |1+a    1     1|
|A| =  | 1    1+b    1|   = abc*(1 + 1/a + 1/b + 1/c) 
       | 1     1   1+c|

a, b, and c do not equal 0.

|A| = (1+a)*[(1+b)*(1+c)-1*1] - 1*[1*(1+c)-1*1] + 1*[1*1-(1+b)*1] =
    = (1+a)*[1+c+b+bc-1] - [1+c-1] + [1-1-b] =
    = (1+a)*(c+b+bc) - c - b =
    = (c+b+bc+ac+ab+abc) - c - b =
    = c + b + bc + ac + ab + abc - c - b =
    = abc + bc + ac + ab =       (since a, b, and c do not equal 0)
    = abc*( abc/abc + bc/abc + ac/abc + ab/abc) =
    = abc*(1 + 1/a + 1/b + 1/c) 


             -----------------------------

3) If A is an idempotent matrix (A^2 = A), prove that the determinant
of A is either 0 or 1.

Recall that if A and B are square matrices of the same size then
det(AB) = det(A)*det(B).

det(A^2) = det(A)^2

Since A^2 = A we have that:
det(A)^2 = det(A^2) = det(A) 

Then:
0 = det(A)^2 - det(A) = det(A)*[det(A) - 1]

The solutions of the above equation are:
det(A) = 0
or
det(A) = 1

           -----------------------------------------

Regarding to the exersice 6), to answer you must check if each axiom
satisfies also the properties of the continuous functions:

For example the function ZERO is in the set, and the sum of two
continuous functions and the multiplication by a scalar do not affect
their continuous condition.

For example 1*f = f for all f in C[0,1], etc.


Wishing the best for you in the finals, best regards.
livioflores-ga

Thanks for the help!

Hi!!

I cannot answer all these questions, but at least I can give you some answers:
Question 1, I really do not understand the statement.
Question 4 and , I cannot figure how to solve them.


Here is what I can do for you:

2) Prove the property.

       |1+a    1     1|
|A| =  | 1    1+b    1|   = abc*(1 + 1/a + 1/b + 1/c) 
       | 1     1   1+c|

a, b, and c do not equal 0.

|A| = (1+a)*[(1+b)*(1+c)-1*1] - 1*[1*(1+c)-1*1] + 1*[1*1-(1+b)*1] =
    = (1+a)*[1+c+b+bc-1] - [1+c-1] + [1-1-b] =
    = (1+a)*(c+b+bc) - c - b =
    = (c+b+bc+ac+ab+abc) - c - b =
    = c + b + bc + ac + ab + abc - c - b =
    = abc + bc + ac + ab =       (since a, b, and c do not equal 0)
    = abc*( abc/abc + bc/abc + ac/abc + ab/abc) =
    = abc*(1 + 1/a + 1/b + 1/c) 


             -----------------------------

3) If A is an idempotent matrix (A^2 = A), prove that the determinant
of A is either 0 or 1.

Recall that if A and B are square matrices of the same size then
det(AB) = det(A)*det(B).

det(A^2) = det(A)^2

Since A^2 = A we have that:
det(A)^2 = det(A^2) = det(A) 

Then:
0 = det(A)^2 - det(A) = det(A)*[det(A) - 1]

The solutions of the above equation are:
det(A) = 0
or
det(A) = 1

           -----------------------------------------

Regarding to the exersice 69, to answer you must check if each axiom
satisfies also the properties of the continuous functions:

For example the function ZERO is in the set, and the sum of two
continuous functions and the multiplication by a scalar do not affect
their continuous condition.

For example 1*f = f for all f in C[0,1], etc.


I hope that this helps you at least a little.
Note that I posted this in the comments section, so you do not be charged for this.

Regards.
livioflores-ga