Some linear algebra problems i with

[skybluekitty]

Let A and B be nxn matrices.
1. Suppose that AB=AC and det A does not equal 0. Show that B=C

2. Show that A is nonsingular if and only if A transpose is nonsingular.

3. Show that det AB = det BA.

4. Show that det AB = 0 if and only if det A=0 or det B=0

5. Show that if AB= -BA and n is odd, then A or B is singular.

6. Show that det A*Atranspose is greater than equal to 0

7. Show that det A*Btranspose = det Atranspose* det B

8. Let A be nxn skew-symmetric matrix. If n is odd, show that det A=0

9. Show that 3x3 vandermonde matrix has a determinant equal to (a-b)(b-c)(c-a) The matrix is
[1 1 1
a b c
a^2 b^2 c^2]
Thank you.

 

[lukaszh]

1. $$AB=AC\Rightarrow A(B-C)=0$$
if A is regular, then there are all pivots nonzero. Then only one way is to satisfy that equation, so $$B-C=0\Rightarrow B=C\qquad\square$$
2. Take any matrix in echelon form, with some pivots. If one of them is zero, then also traspose has a zero pivot. Then A is singular and A transpose is singular.
3. $$\det A\det B=\det A\det B\Rightarrow\det A\det B=\det B\det A\Rightarrow\det AB=\det BA$$

 

[HallsofIvy]

I would really like to see some work on your part. If nothing else it would help to determine what kind of hints would help you. For example, I can see three different ways to do problem 1 but I don't know which way would be best for you.

 

[skybluekitty]

4. Show that det AB = 0 if and only if det A=0 or det B=0
well i know that I have to show two parts for this one
part 1 that assume that det AB=0 then show that det A=0 or B=0
part 2 assume that det A=0 or B=0 then show that det AB = 0
but I have hard time coming up with a good organization and details for this kinds of problem.


5. Show that if AB= -BA and n is odd, then A or B is singular.
i don't have any clue how to start this one... please give me any hints..

6. Show that det A*Atranspose is greater than equal to 0
hmmm i have no clue...
7. Show that det A*Btranspose = det Atranspose* det B

8. Let A be nxn skew-symmetric matrix. If n is odd, show that det A=0

9. Show that 3x3 vandermonde matrix has a determinant equal to (a-b)(b-c)(c-a) The matrix is
[1 1 1
a b c
a^2 b^2 c^2]

when i found the det for this.. I got bc^2+ca^2+ab^2-ba^2-cb^2-ac^2.. i don't know if this is right.. and don't know where to go from there...

I am trying my best and if anyone could give me some type of hints or help me through these problems... that would be great...
Thanks

 

[sutupidmath]

4. You can use the result of a theorem( i don't know whether they expect you to prove it as well or not)

det(AB)=det(A)det(B).

Now if you suppose that det(AB)=0=> det(A)det(B)=0=>...? and vice-versa

5. is n supposed to be the dimension of the matrices A and B?

 

[skybluekitty]

for num 5. yes n is suppose to be the demention of the matrices, so they are square matrices.

 

[sutupidmath]

Also you can, probbaly use another result:

det(A)=det(A^T)

A^T=A transpoze. This will help you for 6 and 7..