LOOSEr Group Title what is a 3x3 invertible matrix? one year ago one year ago

1. tkhunny Group Title

Are you OK with what a 3x3 matrix is?

2. mathsmind Group Title

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

3. LOOSEr Group Title

a solved example would be great

4. mathsmind Group Title

ok

5. LOOSEr Group Title

@mathsmind ???

6. LOOSEr Group Title

which means its determinant cant be 0?

7. mathsmind Group Title

yest i will show you the full steps

8. mathsmind Group Title

consider the following matrix

9. LOOSEr Group Title

PLZ continue

10. mathsmind Group Title

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

11. mathsmind Group Title

Step one: $A^T$

12. mathsmind Group Title

which means transpose of rows and columns of A

13. mathsmind Group Title

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

14. mathsmind Group Title

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

15. LOOSEr Group Title

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

16. mathsmind Group Title

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

17. LOOSEr Group Title

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

18. mathsmind Group Title

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

19. mathsmind Group Title

so we need to find adjA

20. LOOSEr Group Title

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

21. mathsmind Group Title

ok sure

22. LOOSEr Group Title

any idea?

23. mathsmind Group Title

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

24. mathsmind Group Title

i will show you shortly

25. LOOSEr Group Title

@mathsmind ???

26. LOOSEr Group Title

@mathsmind why is taking so long?

27. mathsmind Group Title

sorry back, am at work

28. mathsmind Group Title

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

29. LOOSEr Group Title

yep

30. mathsmind Group Title

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

31. LOOSEr Group Title

But it would be really long

32. mathsmind Group Title

well this is the world of Matrices...

33. mathsmind Group Title

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

34. mathsmind Group Title

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

35. LOOSEr Group Title

Isn't there any short way

36. mathsmind Group Title

have you done hermitian matrices...

37. LOOSEr Group Title

nope

38. mathsmind Group Title

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

39. LOOSEr Group Title

okay

40. mathsmind Group Title

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

41. mathsmind Group Title

see ya tonight

42. LOOSEr Group Title

@ParthKohli

43. Hero Group Title

lol, you still here? I thought you got help

44. LOOSEr Group Title

No

45. LOOSEr Group Title

can u do it?

46. LOOSEr Group Title

Short way

47. ParthKohli Group Title

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

48. ParthKohli Group Title

Nope.

49. yummydum Group Title
50. sauravshakya Group Title

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