N times opeparations(lenear algebra)

[transgalactic]

how to calculate the determinant of this:
http://img6.imageshack.us/img6/4937/16190333pz1.gif

b)
solve this
http://img88.imageshack.us/img88/4556/73223491zz9.th.gif

if it were a defined matrix i could sol it
but there its "n" times

??

 

[transgalactic]

i solved A
how to solve B??
i think of using cramer law
but its an endless equation
??

 

[HallsofIvy]

No, it's not an "endless" equation. For a given n, this is an n+1 by n+1 determinant equal to 0 so it is an nth degree polynomial equation.

Have you tried it for n= 1, n= 2, etc.? You should be able to see a pattern pretty quickly. The roots are 0, 1, 2, ..., n-2.

In fact it should be easy to see that if x= 0, the first two rows are exactly the same so the determinant is 0. If x= 1, the first and third rows are exactly the same. If x= 2, the first and fourth rows are exactly the same, etc.

 

[transgalactic]

ok if i subtract the first row from the rest i get a diagonolized matrix
|1 1 1 .. 1|
|0 -x . . .. 1|
|0 0 1-x 1|
|0 0 0 .. n-2-x|

-x(x-1)(x-2)...((n-2)-x)=0
so the roots are from x=0 to x=n-2
that correct??

 

[HallsofIvy]

Yes, that is correct and a good way to solve it. (You shouldn't have the "1"s above the diagonal though.)