Question

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]]

Answer 1

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

Answer 2

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

Answer 3

Second term being -1

Answer 4

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

Answer 5

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

Answer 6

I guess you can take the determinant?

Answer 7

Then multiply by -1

Answer 8

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

Answer 9

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

Answer 10

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

Answer 11

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

Answer 12

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

Answer 13

**

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

Answer 14

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

Answer 15

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

Answer 16

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

Answer 17

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

Answer 18

the answer is either 0 or 2

0 if you consider 4 terms in the determinant
2 if you consider 24 terms

Answer 19

yahoo answers is not that reliable..

Answer 20

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

Answer 21

they got -40 right ?

Answer 22

Yeah.

Answer 23

remember how to find cofactor of a term ?

Answer 24

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

Answer 25

[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

Answer 26

removing 3rd row and 2nd column,

[3,2,-1]
[-2,1,2]
[-1,2,-1]

Answer 27

And then multiplying all that by -2.

Answer 28

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

Answer 29

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

Answer 30

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

Answer 31

Ohh... My bad..

Answer 32

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

Answer 33

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
```

Answer 34

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

Answer 35

lets work it step by step

Answer 36

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

Answer 37

\[ \begin{vmatrix}a&b\\c&d\\\end{vmatrix} = ad-bc\]

two terms, right ?

Answer 38

Yes.

Answer 39

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