## anonymous one year ago I don't understand this question. Calculate the second term in the cofactor expansion along the third row for the matrix: [[-1,-1,-1,-1],[-1,1,2,0],[1,-1,0,2],[-1,-1,-1,-1]]

1. anonymous

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

2. anonymous

So the third row would be [1, -1, 0, 2]

3. anonymous

Second term being -1

4. anonymous

So do I just use -1 times the rest of the matrix?

5. ganeshie8

$\left[\begin{matrix}-1 & -1 & -1 & -1 \\ -1 & 1 & 2 & 0 \\ 1 & \color{red}{-1} & 0 & 2 \\ -1 & -1 & -1 & -1\end{matrix}\right]$

6. anonymous

I guess you can take the determinant?

7. anonymous

Then multiply by -1

8. anonymous

Take the determinant of the original matrix and times that by -1?

9. ganeshie8

whats the definition of "term" that your professor uses ?

10. ganeshie8

does the determinant has $$4!=24$$ terms or just $$4$$ terms according to your prof ?

11. anonymous

I think she'd say that that'd be just 4 terms.

12. ganeshie8

** then, you just need to find the cofactor of -1, which is at 3,2 location, then multiply that by -1

13. ganeshie8

** cofactor of $$A_{32}$$= $$(-1)^{3+2}\begin{vmatrix}-1&-1&-1\\-1&2&0\\-1&-1&-1\end{vmatrix}$$

14. anonymous

Okay, I am obviously not understanding this question because the answer I got was 0...

15. ganeshie8

Right, I think your prof wants the first term out of those 24 terms in the determinant

16. anonymous

17. anonymous

How did they obtain -20 in this example? https://au.answers.yahoo.com/question/index?qid=20120826032029AANjdNA

18. ganeshie8

the answer is either 0 or 2 0 if you consider 4 terms in the determinant 2 if you consider 24 terms

19. ganeshie8

yahoo answers is not that reliable..

20. anonymous

Aha, I know, I was just trying to work out how they managed to get -20 that's all

21. ganeshie8

they got -40 right ?

22. anonymous

Yeah.

23. ganeshie8

remember how to find cofactor of a term ?

24. anonymous

Yes, but it's a bit hard to write it all on here.

25. ganeshie8

[3,-1,2,-1] [-2,2,1,2] [2,-2,2,1] [-1,3,2,-1] to find the cofactor of -2 at position 3,2 you simply find the determinant of the small matrix obtained by deleting 3rd row and second column, then fix the sign

26. ganeshie8

removing 3rd row and 2nd column, [3,2,-1] [-2,1,2] [-1,2,-1]

27. anonymous

And then multiplying all that by -2.

28. anonymous

What I don't understand is how they ended up getting (-2) x (-20) to get 40.

29. ganeshie8

before that, multiply the determinant by $$(-1)^{3+2}$$ to get the cofactor

30. ganeshie8

what do you get for determinant of [3,2,-1] [-2,1,2] [-1,2,-1]

31. anonymous

32. anonymous

So for my particular question, the determinant of the submatrix is just 0, does this mean that the answer is just 0 then?

33. ganeshie8

let me just quote my previous reply  the answer is either 0 or 2 0 if you consider 4 terms in the determinant 2 if you consider 24 terms 

34. anonymous

Yeah, I remember reading that, I was just confused as to how you managed to get 2 as well if I considered 24 terms rather than 4

35. ganeshie8

lets work it step by step

36. ganeshie8

how many terms are there in the determinant of a 2x2 matrix ?

37. ganeshie8

$\begin{vmatrix}a&b\\c&d\\\end{vmatrix} = ad-bc$ two terms, right ?

38. anonymous

Yes.

39. ganeshie8

how about the determinant of a $$3\times 3$$ matrix, how many terms ?