## liliy Group Title 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 2 years ago

1. myininaya Group Title

What is An?

2. liliy Group Title

the matrix

3. liliy Group Title

commonly seen as Ax=b. this is An=0.

4. myininaya Group Title

Oh I thought there might have been a difference because you called the matrix A then you called it An

5. myininaya Group Title

Are you saying n is an eigenvector ?

6. liliy Group Title

honesly i dont know wat to do. this is what the teacher asked us

7. myininaya Group Title

Or are you say An is the matrix?

8. liliy Group Title

idk..lol

9. myininaya Group Title

Ok I think |dw:1340595284758:dw| and |dw:1340595304816:dw|

10. myininaya Group Title

Is that what you think?

11. myininaya Group Title

But that doesn't make since that A_n would be the matrix with nothing but zero entries

12. myininaya Group Title

do you mean the determinant is 0?

13. myininaya Group Title

|A_n|=0?

14. myininaya Group Title

that would make since

15. myininaya Group Title

sense*

16. myininaya Group Title

I think that is what you mean

17. myininaya Group Title

so do you know how to find the determinant of a matrix?

18. liliy Group Title

19. myininaya Group Title

Try finding the determinant of A_2 ? What do you get?

20. liliy Group Title

zero

21. myininaya Group Title

22. liliy Group Title

same

23. myininaya Group Title

Ok so we have convinced ourselves that |A_n|=0 But we must prove it

24. myininaya Group Title

|dw:1340595692432:dw|

25. myininaya Group Title

I would just show a little work for this show a pattern you know

26. myininaya Group Title

like how you did for A_3 and then do the nth term you know what I mean?

27. liliy Group Title

how do u find determinant for 3x3 or bigger matrix?

28. myininaya Group Title

oh ok for an A_3 |dw:1340595913293:dw|

29. myininaya Group Title

|dw:1340595950507:dw|

30. myininaya Group Title

The signs alternate

31. myininaya Group Title

Like you take top entries

32. liliy Group Title

only the first row?

33. myininaya Group Title

And take everything that isn't below that entry

34. liliy Group Title

like if its 10x10 u still only do the first row /?

35. myininaya Group Title

For |A_4| |dw:1340596049002:dw|

36. myininaya Group Title

|dw:1340596102188:dw|

37. myininaya Group Title

and you already know how to find the determinant for a 3 by 3

38. myininaya Group Title

Same thing just take the top entries and do the signs alternating

39. eliassaab Group Title

$A=\left( \begin{array}{cccc} 0 & 1 & 2 & 3 \\ 0 & 0 & 1 & 4 \\ 0 & 0 & 0 & 3 \\ 0 & 0 & 0 & 0 \\ \end{array} \right)\\ A^2=\left( \begin{array}{cccc} 0 & 0 & 1 & 10 \\ 0 & 0 & 0 & 3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} \right)\\ A^3=\left( \begin{array}{cccc} 0 & 0 & 0 & 3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} \right)\\ A^4=\left( \begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} \right)$

40. liliy Group Title

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

41. myininaya Group Title

yes just like i did above for the 3 by 3

42. myininaya Group Title

But not all the top entries are 0

43. liliy Group Title

@eliassaab i dont really understnad wat u wrote

44. liliy Group Title

right...

45. eliassaab Group Title

See my example below and examine what is going on?

46. liliy Group Title

you have zeros. but what are you doing to the matrix?

47. myininaya Group Title

Do you think he means to raise A to a power @eliassaab ?

48. myininaya Group Title

Instead of finding the determinant ?

49. eliassaab Group Title

You raise it to the power 2, then 3, then 4.

50. myininaya Group Title

Ok I'm sorry @liliy I don't know what your question is asking anymore.

51. eliassaab Group Title

You do not need to deal with determina

52. eliassaab Group Title

determinant

53. liliy Group Title

so can you start over with me?

54. liliy Group Title

what does a^n=0 even mean?

55. eliassaab Group Title

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.

56. eliassaab Group Title

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.

57. myininaya Group Title

So you are just giving another way right @eliassaab Do you think I interpreted is question correctly?

58. eliassaab Group Title

@myininaya, you do not need determinant to do that,

59. myininaya Group Title

Yeah I know, but I'm asking you if I interpreted it correctly?

60. liliy Group Title

@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..

61. eliassaab Group Title

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 $$f(A)=A^n=0$$