anonymous 3 years ago what is a 3x3 invertible matrix?

1. tkhunny

Are you OK with what a 3x3 matrix is?

2. anonymous

do you want a formal definition or a solved example showing all steps of inverting a 3x3 matrix?

3. anonymous

a solved example would be great

4. anonymous

ok

5. anonymous

@mathsmind ???

6. anonymous

which means its determinant cant be 0?

7. anonymous

yest i will show you the full steps

8. anonymous

consider the following matrix

9. anonymous

PLZ continue

10. anonymous

$A=\left[\begin{matrix}1 & -2 & 0\\ 3& 1 & 5\\ -1 & 2 & 3\end{matrix}\right]$

11. anonymous

Step one: $A^T$

12. anonymous

which means transpose of rows and columns of A

13. anonymous

$A^T=\left[\begin{matrix}1 & 3 & -1\\ -2& 1 & 2\\ 0 & 5 & 3\end{matrix}\right]$

14. anonymous

Step Two: find the adjoint or adjugate of A, which implies that we replace each element by their cofactor...

15. anonymous

If P = \begin{bmatrix} a &b &c \\ d &e &f \\ g &h &i \end{bmatrix}

16. anonymous

now please concentrate on this step it is simple but requires a bit of accuracy

17. anonymous

And P^4=2P how can we find P

18. anonymous

let's finish this first part then i will come to the 2nd question, you asked me first for a numerical example

19. anonymous

so we need to find adjA

20. anonymous

I know how to find a inverse matrix. So, can u plz do the 2nd question

21. anonymous

ok sure

22. anonymous

any idea?

23. anonymous

yes sorry am on the phone, but just a quick hint for P^n we use the principle of induction

24. anonymous

i will show you shortly

25. anonymous

@mathsmind ???

26. anonymous

@mathsmind why is taking so long?

27. anonymous

sorry back, am at work

28. anonymous

now P^4 means you are multiplying your matrix by itself 4 times.

29. anonymous

yep

30. anonymous

you know how to multiply the, it would be PP=P^2, then P^2P=P^3. then P^3P=P^4

31. anonymous

But it would be really long

32. anonymous

well this is the world of Matrices...

33. anonymous

are you thinking of algebraic operation on matrices such as taking the inverse of 2P

34. anonymous

or subtracting p^4-2P=0 in order to find the new P?

35. anonymous

Isn't there any short way

36. anonymous

have you done hermitian matrices...

37. anonymous

nope

38. anonymous

listen can we kindly finish this tonight because it is 5 am and i need to go ...

39. anonymous

okay

40. anonymous

i will provide you with full solution even if you are not online ok

41. anonymous

see ya tonight

42. anonymous

@ParthKohli

43. Hero

lol, you still here? I thought you got help

44. anonymous

No

45. anonymous

can u do it?

46. anonymous

Short way

47. ParthKohli

I haven't done matrices yet: I am stuck with number theory at the moment.

48. ParthKohli

Nope.

49. anonymous
50. anonymous

@looser there are many matrices such that P^4=2P but only one is invertible which is |dw:1361026643638:dw|