(An)i,j = {1, i = j-1
{1, i = j+1
{10, i = j-2d
{0, otherwise
Example for A_5 in the picture.
We will define dn = det(An)
Find the withdawal formula for dn.
#For every n in N, we will define An in R(nxn) in the following way:
flint quarry
rough pebbleBOT
(An)i,j = {1, i = j-1
{1, i = j+1
{10, i = j-2d
...
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real moss
(An)i,j = {1, i = j-1
{1, i = j+1
{10, i = j-2d
...
you mean to compute dn ?
flint quarry
what do you mean compute it?
real moss
express dn in terms of n
real moss
do you mean the characteristic polynomial of An ?
flint quarry
It is when you express a term in a series (in this case dn) by previous terms. For example the withdrawal function of the fibonnaci numbers is Fn = F(n-1) + F(n-2).
I might have gotten the term wrong in english though
sage crag
do you have small examples?
flint quarry
I think you might know it as Recurrence relation
flint quarry
do you have small examples?
to what?
sage crag
of a withdrawal function
flint quarry
For example the withdrawal function of the fibonnaci numbers is Fn = F(n-1) + F(n-2) for n>=2.
sage crag
is it reasonable to speak of a withdrawal function of a 3x3 determinant?
oh, it's the recursive relationship itself
flint quarry
Yes
real moss
ok, you want to find a recursive relation for dn
flint quarry
In this question, I have tried to open the matrix from the top row
real moss
do you know tridiagonal determinants ?
flint quarry
No, what is that?
real moss
they are determinants of tridiagonal matrices https://en.wikipedia.org/wiki/Tridiagonal_matrix
they are easy to expand, and when the coefficients of each diagonal are constant you obtain easily a recurrence relation
now, An is not a tridiagonal matrix
however, using transvections, you can express det An as the determinant of a tridiagonal matrix
(by removing the 10's)
(using column operations)
real moss
for A5, perform the operations C3<-C3-10C2, C4<-C4-10C3 and C5<-C5-10C4
which matrix do you obtain ?
(nvm, this one doesn't work)
flint quarry
why not though?
real moss
well because at some point you create a coefficient 101
which needs to be used to cancel the 10
flint quarry
Ok
I thought we might wanna try oppening the matrix from the top row
because if you remove R1, C1, we get d(n-1)
real moss
times 0
real moss
yes
real moss
the first determinant is easy to compute directly
the second determinant, you can express it in terms of Dm for some m
flint quarry
You mean the one we multiply by 10?
real moss
yes
flint quarry
By using column operations?
flint quarry
how?
real moss
you recognize a block diagonal matrix
willow daggerBOT
@flint quarry
willow daggerBOT
@real moss @sage crag The user still needs help with this help request.
flint quarry
+close