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 2 years 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?
 2 years 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?

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liliy
 2 years ago
Best ResponseYou've already chosen the best response.0commonly seen as Ax=b. this is An=0.

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

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.0Are you saying n is an eigenvector ?

liliy
 2 years ago
Best ResponseYou've already chosen the best response.0honesly i dont know wat to do. this is what the teacher asked us

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.0Or are you say An is the matrix?

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.0Ok I think dw:1340595284758:dw and dw:1340595304816:dw

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.0Is that what you think?

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.0But that doesn't make since that A_n would be the matrix with nothing but zero entries

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.0do you mean the determinant is 0?

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.0that would make since

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.0I think that is what you mean

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.0so do you know how to find the determinant of a matrix?

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.0Try finding the determinant of A_2 ? What do you get?

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.0Ok so we have convinced ourselves that A_n=0 But we must prove it

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.0dw:1340595692432:dw

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.0I would just show a little work for this show a pattern you know

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.0like how you did for A_3 and then do the nth term you know what I mean?

liliy
 2 years ago
Best ResponseYou've already chosen the best response.0how do u find determinant for 3x3 or bigger matrix?

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.0oh ok for an A_3 dw:1340595913293:dw

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.0dw:1340595950507:dw

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.0Like you take top entries

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.0And take everything that isn't below that entry

liliy
 2 years ago
Best ResponseYou've already chosen the best response.0like if its 10x10 u still only do the first row /?

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.0For A_4 dw:1340596049002:dw

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.0dw:1340596102188:dw

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.0and you already know how to find the determinant for a 3 by 3

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.0Same thing just take the top entries and do the signs alternating

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

liliy
 2 years ago
Best ResponseYou've already chosen the best response.0but 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

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.0yes just like i did above for the 3 by 3

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.0But not all the top entries are 0

liliy
 2 years ago
Best ResponseYou've already chosen the best response.0@eliassaab i dont really understnad wat u wrote

eliassaab
 2 years ago
Best ResponseYou've already chosen the best response.1See my example below and examine what is going on?

liliy
 2 years ago
Best ResponseYou've already chosen the best response.0you have zeros. but what are you doing to the matrix?

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.0Do you think he means to raise A to a power @eliassaab ?

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.0Instead of finding the determinant ?

eliassaab
 2 years ago
Best ResponseYou've already chosen the best response.1You raise it to the power 2, then 3, then 4.

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.0Ok I'm sorry @liliy I don't know what your question is asking anymore.

eliassaab
 2 years ago
Best ResponseYou've already chosen the best response.1You do not need to deal with determina

liliy
 2 years ago
Best ResponseYou've already chosen the best response.0so can you start over with me?

liliy
 2 years ago
Best ResponseYou've already chosen the best response.0what does a^n=0 even mean?

eliassaab
 2 years ago
Best ResponseYou've already chosen the best response.1Any 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.

eliassaab
 2 years ago
Best ResponseYou've already chosen the best response.1Look 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.

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

eliassaab
 2 years ago
Best ResponseYou've already chosen the best response.1@myininaya, you do not need determinant to do that,

myininaya
 2 years ago
Best ResponseYou've already chosen the best response.0Yeah I know, but I'm asking you if I interpreted it correctly?

liliy
 2 years ago
Best 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..

eliassaab
 2 years ago
Best ResponseYou've already chosen the best response.1Here 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 \(f(A)=A^n=0\)
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