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

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1. tkhunny

Are you OK with what a 3x3 matrix is?

2. mathsmind

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

3. LOOSEr

a solved example would be great

4. mathsmind

ok

5. LOOSEr

@mathsmind ???

6. LOOSEr

which means its determinant cant be 0?

7. mathsmind

yest i will show you the full steps

8. mathsmind

consider the following matrix

9. LOOSEr

PLZ continue

10. mathsmind

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

11. mathsmind

Step one: $A^T$

12. mathsmind

which means transpose of rows and columns of A

13. mathsmind

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

14. mathsmind

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

15. LOOSEr

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

16. mathsmind

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

17. LOOSEr

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

18. mathsmind

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

19. mathsmind

so we need to find adjA

20. LOOSEr

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

21. mathsmind

ok sure

22. LOOSEr

any idea?

23. mathsmind

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

24. mathsmind

i will show you shortly

25. LOOSEr

@mathsmind ???

26. LOOSEr

@mathsmind why is taking so long?

27. mathsmind

sorry back, am at work

28. mathsmind

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

29. LOOSEr

yep

30. mathsmind

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

31. LOOSEr

But it would be really long

32. mathsmind

well this is the world of Matrices...

33. mathsmind

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

34. mathsmind

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

35. LOOSEr

Isn't there any short way

36. mathsmind

have you done hermitian matrices...

37. LOOSEr

nope

38. mathsmind

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

39. LOOSEr

okay

40. mathsmind

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

41. mathsmind

see ya tonight

42. LOOSEr

@ParthKohli

43. Hero

lol, you still here? I thought you got help

44. LOOSEr

No

45. LOOSEr

can u do it?

46. LOOSEr

Short way

47. ParthKohli

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

48. ParthKohli

Nope.

49. yummydum
50. sauravshakya

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