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