anonymous
  • anonymous
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?
Mathematics
  • Stacey Warren - Expert brainly.com
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chestercat
  • chestercat
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myininaya
  • myininaya
What is An?
anonymous
  • anonymous
the matrix
anonymous
  • anonymous
commonly seen as Ax=b. this is An=0.

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myininaya
  • myininaya
Oh I thought there might have been a difference because you called the matrix A then you called it An
myininaya
  • myininaya
Are you saying n is an eigenvector ?
anonymous
  • anonymous
honesly i dont know wat to do. this is what the teacher asked us
myininaya
  • myininaya
Or are you say An is the matrix?
anonymous
  • anonymous
idk..lol
myininaya
  • myininaya
Ok I think |dw:1340595284758:dw| and |dw:1340595304816:dw|
myininaya
  • myininaya
Is that what you think?
myininaya
  • myininaya
But that doesn't make since that A_n would be the matrix with nothing but zero entries
myininaya
  • myininaya
do you mean the determinant is 0?
myininaya
  • myininaya
|A_n|=0?
myininaya
  • myininaya
that would make since
myininaya
  • myininaya
sense*
myininaya
  • myininaya
I think that is what you mean
myininaya
  • myininaya
so do you know how to find the determinant of a matrix?
anonymous
  • anonymous
ya ad-bc
myininaya
  • myininaya
Try finding the determinant of A_2 ? What do you get?
anonymous
  • anonymous
zero
myininaya
  • myininaya
Ok what about A_3
anonymous
  • anonymous
same
myininaya
  • myininaya
Ok so we have convinced ourselves that |A_n|=0 But we must prove it
myininaya
  • myininaya
|dw:1340595692432:dw|
myininaya
  • myininaya
I would just show a little work for this show a pattern you know
myininaya
  • myininaya
like how you did for A_3 and then do the nth term you know what I mean?
anonymous
  • anonymous
how do u find determinant for 3x3 or bigger matrix?
myininaya
  • myininaya
oh ok for an A_3 |dw:1340595913293:dw|
myininaya
  • myininaya
|dw:1340595950507:dw|
myininaya
  • myininaya
The signs alternate
myininaya
  • myininaya
Like you take top entries
anonymous
  • anonymous
only the first row?
myininaya
  • myininaya
And take everything that isn't below that entry
anonymous
  • anonymous
like if its 10x10 u still only do the first row /?
myininaya
  • myininaya
For |A_4| |dw:1340596049002:dw|
myininaya
  • myininaya
|dw:1340596102188:dw|
myininaya
  • myininaya
and you already know how to find the determinant for a 3 by 3
myininaya
  • myininaya
Same thing just take the top entries and do the signs alternating
anonymous
  • anonymous
\[ A=\left( \begin{array}{cccc} 0 & 1 & 2 & 3 \\ 0 & 0 & 1 & 4 \\ 0 & 0 & 0 & 3 \\ 0 & 0 & 0 & 0 \\ \end{array} \right)\\ A^2=\left( \begin{array}{cccc} 0 & 0 & 1 & 10 \\ 0 & 0 & 0 & 3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} \right)\\ A^3=\left( \begin{array}{cccc} 0 & 0 & 0 & 3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} \right)\\ A^4=\left( \begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} \right) \]
anonymous
  • anonymous
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
myininaya
  • myininaya
yes just like i did above for the 3 by 3
myininaya
  • myininaya
But not all the top entries are 0
anonymous
  • anonymous
@eliassaab i dont really understnad wat u wrote
anonymous
  • anonymous
right...
anonymous
  • anonymous
See my example below and examine what is going on?
anonymous
  • anonymous
you have zeros. but what are you doing to the matrix?
myininaya
  • myininaya
Do you think he means to raise A to a power @eliassaab ?
myininaya
  • myininaya
Instead of finding the determinant ?
anonymous
  • anonymous
You raise it to the power 2, then 3, then 4.
myininaya
  • myininaya
Ok I'm sorry @liliy I don't know what your question is asking anymore.
anonymous
  • anonymous
You do not need to deal with determina
anonymous
  • anonymous
determinant
anonymous
  • anonymous
so can you start over with me?
anonymous
  • anonymous
what does a^n=0 even mean?
anonymous
  • anonymous
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.
anonymous
  • anonymous
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.
myininaya
  • myininaya
So you are just giving another way right @eliassaab Do you think I interpreted is question correctly?
anonymous
  • anonymous
@myininaya, you do not need determinant to do that,
myininaya
  • myininaya
Yeah I know, but I'm asking you if I interpreted it correctly?
anonymous
  • anonymous
@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..
anonymous
  • anonymous
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 \(f(A)=A^n=0\)

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