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Show that if A is nxn and has all 0’s on and below the diagonal then An = 0. Hint: do not at first be too ambitious. First find A2 and observe something useful about it. What about A3?
 one year ago
 one year ago
Show that if A is nxn and has all 0’s on and below the diagonal then An = 0. Hint: do not at first be too ambitious. First find A2 and observe something useful about it. What about A3?
 one year ago
 one year ago

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liliyBest ResponseYou've already chosen the best response.0
commonly seen as Ax=b. this is An=0.
 one year ago

myininayaBest ResponseYou've already chosen the best response.0
Oh I thought there might have been a difference because you called the matrix A then you called it An
 one year ago

myininayaBest ResponseYou've already chosen the best response.0
Are you saying n is an eigenvector ?
 one year ago

liliyBest ResponseYou've already chosen the best response.0
honesly i dont know wat to do. this is what the teacher asked us
 one year ago

myininayaBest ResponseYou've already chosen the best response.0
Or are you say An is the matrix?
 one year ago

myininayaBest ResponseYou've already chosen the best response.0
Ok I think dw:1340595284758:dw and dw:1340595304816:dw
 one year ago

myininayaBest ResponseYou've already chosen the best response.0
Is that what you think?
 one year ago

myininayaBest ResponseYou've already chosen the best response.0
But that doesn't make since that A_n would be the matrix with nothing but zero entries
 one year ago

myininayaBest ResponseYou've already chosen the best response.0
do you mean the determinant is 0?
 one year ago

myininayaBest ResponseYou've already chosen the best response.0
that would make since
 one year ago

myininayaBest ResponseYou've already chosen the best response.0
I think that is what you mean
 one year ago

myininayaBest ResponseYou've already chosen the best response.0
so do you know how to find the determinant of a matrix?
 one year ago

myininayaBest ResponseYou've already chosen the best response.0
Try finding the determinant of A_2 ? What do you get?
 one year ago

myininayaBest ResponseYou've already chosen the best response.0
Ok so we have convinced ourselves that A_n=0 But we must prove it
 one year ago

myininayaBest ResponseYou've already chosen the best response.0
dw:1340595692432:dw
 one year ago

myininayaBest ResponseYou've already chosen the best response.0
I would just show a little work for this show a pattern you know
 one year ago

myininayaBest ResponseYou've already chosen the best response.0
like how you did for A_3 and then do the nth term you know what I mean?
 one year ago

liliyBest ResponseYou've already chosen the best response.0
how do u find determinant for 3x3 or bigger matrix?
 one year ago

myininayaBest ResponseYou've already chosen the best response.0
oh ok for an A_3 dw:1340595913293:dw
 one year ago

myininayaBest ResponseYou've already chosen the best response.0
dw:1340595950507:dw
 one year ago

myininayaBest ResponseYou've already chosen the best response.0
Like you take top entries
 one year ago

myininayaBest ResponseYou've already chosen the best response.0
And take everything that isn't below that entry
 one year ago

liliyBest ResponseYou've already chosen the best response.0
like if its 10x10 u still only do the first row /?
 one year ago

myininayaBest ResponseYou've already chosen the best response.0
For A_4 dw:1340596049002:dw
 one year ago

myininayaBest ResponseYou've already chosen the best response.0
dw:1340596102188:dw
 one year ago

myininayaBest ResponseYou've already chosen the best response.0
and you already know how to find the determinant for a 3 by 3
 one year ago

myininayaBest ResponseYou've already chosen the best response.0
Same thing just take the top entries and do the signs alternating
 one year ago

eliassaabBest ResponseYou've already chosen the best response.1
\[ A=\left( \begin{array}{cccc} 0 & 1 & 2 & 3 \\ 0 & 0 & 1 & 4 \\ 0 & 0 & 0 & 3 \\ 0 & 0 & 0 & 0 \\ \end{array} \right)\\ A^2=\left( \begin{array}{cccc} 0 & 0 & 1 & 10 \\ 0 & 0 & 0 & 3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} \right)\\ A^3=\left( \begin{array}{cccc} 0 & 0 & 0 & 3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} \right)\\ A^4=\left( \begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} \right) \]
 one year ago

liliyBest ResponseYou've already chosen the best response.0
but the determinant of the new 3x3 is gonna also be broken down right? .. im ur case its zero bec the coefficient is zero so it odsnt really mater
 one year ago

myininayaBest ResponseYou've already chosen the best response.0
yes just like i did above for the 3 by 3
 one year ago

myininayaBest ResponseYou've already chosen the best response.0
But not all the top entries are 0
 one year ago

liliyBest ResponseYou've already chosen the best response.0
@eliassaab i dont really understnad wat u wrote
 one year ago

eliassaabBest ResponseYou've already chosen the best response.1
See my example below and examine what is going on?
 one year ago

liliyBest ResponseYou've already chosen the best response.0
you have zeros. but what are you doing to the matrix?
 one year ago

myininayaBest ResponseYou've already chosen the best response.0
Do you think he means to raise A to a power @eliassaab ?
 one year ago

myininayaBest ResponseYou've already chosen the best response.0
Instead of finding the determinant ?
 one year ago

eliassaabBest ResponseYou've already chosen the best response.1
You raise it to the power 2, then 3, then 4.
 one year ago

myininayaBest ResponseYou've already chosen the best response.0
Ok I'm sorry @liliy I don't know what your question is asking anymore.
 one year ago

eliassaabBest ResponseYou've already chosen the best response.1
You do not need to deal with determina
 one year ago

liliyBest ResponseYou've already chosen the best response.0
so can you start over with me?
 one year ago

liliyBest ResponseYou've already chosen the best response.0
what does a^n=0 even mean?
 one year ago

eliassaabBest ResponseYou've already chosen the best response.1
Any matrix like yours, when you raise it to the power 2, you get what is first above the diagonal is zero When you raise it to the power 2, you get the first and the second above the diagonal to be zero. When you raise it to the power 3, you get the first and the second and third above the diagonal to be zero. When you raise it to the power 4, you get everything zero.
 one year ago

eliassaabBest ResponseYou've already chosen the best response.1
Look at A^4 in my example above to see that A^4=0, this means all the entries of the matrix A^4 are zeros.
 one year ago

myininayaBest ResponseYou've already chosen the best response.0
So you are just giving another way right @eliassaab Do you think I interpreted is question correctly?
 one year ago

eliassaabBest ResponseYou've already chosen the best response.1
@myininaya, you do not need determinant to do that,
 one year ago

myininayaBest ResponseYou've already chosen the best response.0
Yeah I know, but I'm asking you if I interpreted it correctly?
 one year ago

liliyBest ResponseYou've already chosen the best response.0
@eliassaab i dont undesrtnad how you started to do the problem. my teacher said start with a^2 .. and move to bigger ones... so wat is a= to a 4x4 and then writing a^2... a^3..
 one year ago

eliassaabBest ResponseYou've already chosen the best response.1
Here is a quick proof using the characteristic polynomial f(x) of the matrix A that says the f(A)=0. Our matrix has\( f(x)=x^n\), hence \(f(A)=A^n=0\)
 one year ago
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