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]]

- Clarence

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- Clarence

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

- Clarence

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

- Clarence

Second term being -1

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## More answers

- Clarence

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

- ganeshie8

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

- anonymous

I guess you can take the determinant?

- anonymous

Then multiply by -1

- Clarence

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

- ganeshie8

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

- ganeshie8

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

- Clarence

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

- ganeshie8

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

- ganeshie8

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

- Clarence

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

- ganeshie8

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

- Clarence

There's a similar question on this website that went unanswered:http://openstudy.com/study#/updates/5039e2afe4b043c156a3277c

- Clarence

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

- ganeshie8

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

- ganeshie8

yahoo answers is not that reliable..

- Clarence

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

- ganeshie8

they got -40 right ?

- Clarence

Yeah.

- ganeshie8

remember how to find cofactor of a term ?

- Clarence

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

- 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

- ganeshie8

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

- Clarence

And then multiplying all that by -2.

- Clarence

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

- ganeshie8

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

- ganeshie8

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

- Clarence

Ohh... My bad..

- Clarence

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

- 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
```

- Clarence

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

- ganeshie8

lets work it step by step

- ganeshie8

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

- ganeshie8

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

- Clarence

Yes.

- ganeshie8

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

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