Efficiently Calculating Determinants Using Cofactors and Expansion

  • Thread starter lo2
  • Start date
In summary, the conversation discusses the expansion of a determinant using the notation of summation. The summation expands to the long hand version and is equal to the given expression. The concept of determinants is mentioned but is not relevant to the conversation. The original poster may have intended to ask about the meaning of A_{ij}, but it is unclear. They also express confusion about the sentence structure and ask for the post to be deleted.
  • #1
lo2
Can this be written.

[tex]
\[
\det{\textbf{A}}=a_{11}a'_{11}+a_{12}a'_{12}+\ldots+a_{1n}a'{1n}
\]

[/tex]

As. (Where I just use p instead of n for obvious reasons)

[tex]
\[
\sum^{n}_{p=1}{a_{1p}a'_{1p}}
\]
[/tex]

Since.

[tex]
\[
a'_{ij}=(-1)^{i+j}(\det{\textbf{A}_{ij}})
\]
[/tex]

And yes we are talking determinants.
 
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  • #2
Come on please help, would be most appreciated.
 
  • #3
If you want help then write in a clear manner that people can understand, preferably in complete sentences - if people have to work hard at understanding what the question is they will tend not to bother working out what you intend.

What is a'_{rs}? What are you trying to prove?

I suspect that that doesn't matter. If all you're asking is does that summation expand to give the long hand version, then the answer is 'yes'. Just write down what the summation means.

Det has nothing to do with it as far as I can tell. But because you made a big point of saying it was abuot Det I have no idea if I've answered the question you intended to ask.
 
  • #4
lo2 said:
[tex]a_{11}a'_{11}+a_{12}a'_{12}+\ldots+a_{1n}a'_{1n} = \sum^{n}_{p=1}{a_{1p}a'_{1p}} [/tex]

yes, these are equal.

Note, I cleaned up your latex. There is no need to include a \[ inside tex tags - I don't think it does anything.
 
  • #5
I do think I made myself very clear, but well anyway thank you for the help!
 
  • #6
Could this post please be deleted.
 
  • #7
You can delete it yourself.

If you think you were clear, can I ask what you think A_{ij} is? 'Cos we don't have a clue. And the 'since' is completely misleading. As is the sentence structure ('As.' is not a sentence, and is confusingly, given the post, like a plural form of A).
 
  • #8
matt grime said:
You can delete it yourself.

How to do that?
 
  • #9
It's too late now, but next time click on 'edit' and there is an option to delete. The edit option vanishes after some amount of time.
 
  • #10
But I would really like to get this post deleted.
 
  • #11
Why? (added junk to meet minimum post requirement)
 

FAQ: Efficiently Calculating Determinants Using Cofactors and Expansion

How do you calculate a determinant using cofactors and expansion?

To calculate a determinant using cofactors and expansion, you first need to find the size of the matrix. Then, choose a row or column to expand from and calculate the minor of each element in that row or column. Next, multiply each minor by its corresponding cofactor and add all of the resulting products. This will give you the value of the determinant.

What are cofactors?

Cofactors are values that are used in the calculation of determinants. They are found by taking the determinant of the minor of a particular element in a matrix and multiplying it by -1 raised to the power of the sum of its row and column indices.

What is expansion by minors?

Expansion by minors is a method used to calculate determinants. It involves expanding a determinant by choosing a row or column and calculating the minor of each element in that row or column. The minor is then multiplied by its corresponding cofactor and added to the other resulting products to get the value of the determinant.

How is using cofactors and expansion more efficient than other methods?

Using cofactors and expansion is more efficient than other methods because it reduces the number of calculations required. It also allows for the calculation of larger determinants without having to use complex formulas or algorithms.

What are some applications of efficiently calculating determinants using cofactors and expansion?

Efficiently calculating determinants using cofactors and expansion is used in various fields such as linear algebra, physics, and engineering. It can be used to solve systems of linear equations, find eigenvalues and eigenvectors, and determine the invertibility of a matrix, among other applications.

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