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]]

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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]]

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\[\left[\begin{matrix}-1 & -1 & -1 & -1 \\ -1 & 1 & 2 & 0 \\ 1 & -1 & 0 & 2 \\ -1 & -1 & -1 & -1\end{matrix}\right]\]
So the third row would be [1, -1, 0, 2]
Second term being -1

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So do I just use -1 times the rest of the matrix?
\[\left[\begin{matrix}-1 & -1 & -1 & -1 \\ -1 & 1 & 2 & 0 \\ 1 & \color{red}{-1} & 0 & 2 \\ -1 & -1 & -1 & -1\end{matrix}\right]\]
I guess you can take the determinant?
Then multiply by -1
Take the determinant of the original matrix and times that by -1?
whats the definition of "term" that your professor uses ?
does the determinant has \(4!=24\) terms or just \(4\) terms according to your prof ?
I think she'd say that that'd be just 4 terms.
** then, you just need to find the cofactor of -1, which is at 3,2 location, then multiply that by -1
** cofactor of \(A_{32}\)= \((-1)^{3+2}\begin{vmatrix}-1&-1&-1\\-1&2&0\\-1&-1&-1\end{vmatrix}\)
Okay, I am obviously not understanding this question because the answer I got was 0...
Right, I think your prof wants the first term out of those 24 terms in the determinant
There's a similar question on this website that went unanswered:http://openstudy.com/study#/updates/5039e2afe4b043c156a3277c
How did they obtain -20 in this example? https://au.answers.yahoo.com/question/index?qid=20120826032029AANjdNA
the answer is either 0 or 2 0 if you consider 4 terms in the determinant 2 if you consider 24 terms
yahoo answers is not that reliable..
Aha, I know, I was just trying to work out how they managed to get -20 that's all
they got -40 right ?
Yeah.
remember how to find cofactor of a term ?
Yes, but it's a bit hard to write it all on here.
[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
removing 3rd row and 2nd column, [3,2,-1] [-2,1,2] [-1,2,-1]
And then multiplying all that by -2.
What I don't understand is how they ended up getting (-2) x (-20) to get 40.
before that, multiply the determinant by \((-1)^{3+2}\) to get the cofactor
what do you get for determinant of [3,2,-1] [-2,1,2] [-1,2,-1]
Ohh... My bad..
So for my particular question, the determinant of the submatrix is just 0, does this mean that the answer is just 0 then?
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 ```
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
lets work it step by step
how many terms are there in the determinant of a 2x2 matrix ?
\[ \begin{vmatrix}a&b\\c&d\\\end{vmatrix} = ad-bc\] two terms, right ?
Yes.
how about the determinant of a \(3\times 3\) matrix, how many terms ?

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