Determinant of a matrix. Why is it called an expansion of cofactors?

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Why is the word expansion used? What does it mean here?



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  • It seems, from the way the thing is phrased, that the author has already said somewhere before this that the quantities $C_ij$ are called cofactors (and what they are in terms of $A$).
    – Saucy O'Path
    Aug 6 at 17:51











  • I've never thought about the meaning of the word expansion in this context. It probably comes from a loose analogy with expanding brackets, or a binomial expansion... in this case, it means you are expressing the determinant as a sum involving cofactors
    – Calvin Khor
    Aug 6 at 19:11















up vote
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down vote

favorite












Why is the word expansion used? What does it mean here?



enter image description here




linear-algebra matrices determinant




share|cite|improve this question





















  • It seems, from the way the thing is phrased, that the author has already said somewhere before this that the quantities $C_ij$ are called cofactors (and what they are in terms of $A$).
    – Saucy O'Path
    Aug 6 at 17:51











  • I've never thought about the meaning of the word expansion in this context. It probably comes from a loose analogy with expanding brackets, or a binomial expansion... in this case, it means you are expressing the determinant as a sum involving cofactors
    – Calvin Khor
    Aug 6 at 19:11













up vote
1
down vote

favorite









up vote
1
down vote

favorite











Why is the word expansion used? What does it mean here?



enter image description here




linear-algebra matrices determinant




share|cite|improve this question













Why is the word expansion used? What does it mean here?



enter image description here





linear-algebra matrices determinant





share|cite|improve this question












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edited Aug 6 at 18:06









Rodrigo de Azevedo

12.6k41751




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asked Aug 6 at 17:41









Jwan622

1,61211224




1,61211224











  • It seems, from the way the thing is phrased, that the author has already said somewhere before this that the quantities $C_ij$ are called cofactors (and what they are in terms of $A$).
    – Saucy O'Path
    Aug 6 at 17:51











  • I've never thought about the meaning of the word expansion in this context. It probably comes from a loose analogy with expanding brackets, or a binomial expansion... in this case, it means you are expressing the determinant as a sum involving cofactors
    – Calvin Khor
    Aug 6 at 19:11

















  • It seems, from the way the thing is phrased, that the author has already said somewhere before this that the quantities $C_ij$ are called cofactors (and what they are in terms of $A$).
    – Saucy O'Path
    Aug 6 at 17:51











  • I've never thought about the meaning of the word expansion in this context. It probably comes from a loose analogy with expanding brackets, or a binomial expansion... in this case, it means you are expressing the determinant as a sum involving cofactors
    – Calvin Khor
    Aug 6 at 19:11
















It seems, from the way the thing is phrased, that the author has already said somewhere before this that the quantities $C_ij$ are called cofactors (and what they are in terms of $A$).
– Saucy O'Path
Aug 6 at 17:51





It seems, from the way the thing is phrased, that the author has already said somewhere before this that the quantities $C_ij$ are called cofactors (and what they are in terms of $A$).
– Saucy O'Path
Aug 6 at 17:51













I've never thought about the meaning of the word expansion in this context. It probably comes from a loose analogy with expanding brackets, or a binomial expansion... in this case, it means you are expressing the determinant as a sum involving cofactors
– Calvin Khor
Aug 6 at 19:11





I've never thought about the meaning of the word expansion in this context. It probably comes from a loose analogy with expanding brackets, or a binomial expansion... in this case, it means you are expressing the determinant as a sum involving cofactors
– Calvin Khor
Aug 6 at 19:11











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Frequently it happens that in a set of objects that can be composed together in some way, there are objects that are for some reason simpler than others.



In this circumstances it can be convenient to be able to find how simpler objects combine to give an object.



Once found, one can say that the object is decomposed over the simpler objects, but no insight into how simpler objects compose is given.



If the composition is a sum of the simpler objects each multiplied by a factor then one can use the espression "the object is expanded by the simpler objects".
(See e.g. vector expansion, Taylor or Fourier expansion)



If the composition is a product of the simpler objects each raised to some power then one can use the espression "the object is factored into the simpler objects".
(See e.g. integer factorization or polynomial factorization)



Cofactor = complement of factor



$C_11$ is the complement of the factor $a_11$. It is the complement because it is computed by complementation of $a_11$ w.r.t. the matrix, that is, by eliminating by the matrix the row and the column containing $a_11$.






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    Frequently it happens that in a set of objects that can be composed together in some way, there are objects that are for some reason simpler than others.



    In this circumstances it can be convenient to be able to find how simpler objects combine to give an object.



    Once found, one can say that the object is decomposed over the simpler objects, but no insight into how simpler objects compose is given.



    If the composition is a sum of the simpler objects each multiplied by a factor then one can use the espression "the object is expanded by the simpler objects".
    (See e.g. vector expansion, Taylor or Fourier expansion)



    If the composition is a product of the simpler objects each raised to some power then one can use the espression "the object is factored into the simpler objects".
    (See e.g. integer factorization or polynomial factorization)



    Cofactor = complement of factor



    $C_11$ is the complement of the factor $a_11$. It is the complement because it is computed by complementation of $a_11$ w.r.t. the matrix, that is, by eliminating by the matrix the row and the column containing $a_11$.






    share|cite|improve this answer

























      up vote
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      down vote













      Frequently it happens that in a set of objects that can be composed together in some way, there are objects that are for some reason simpler than others.



      In this circumstances it can be convenient to be able to find how simpler objects combine to give an object.



      Once found, one can say that the object is decomposed over the simpler objects, but no insight into how simpler objects compose is given.



      If the composition is a sum of the simpler objects each multiplied by a factor then one can use the espression "the object is expanded by the simpler objects".
      (See e.g. vector expansion, Taylor or Fourier expansion)



      If the composition is a product of the simpler objects each raised to some power then one can use the espression "the object is factored into the simpler objects".
      (See e.g. integer factorization or polynomial factorization)



      Cofactor = complement of factor



      $C_11$ is the complement of the factor $a_11$. It is the complement because it is computed by complementation of $a_11$ w.r.t. the matrix, that is, by eliminating by the matrix the row and the column containing $a_11$.






      share|cite|improve this answer























        up vote
        1
        down vote










        up vote
        1
        down vote









        Frequently it happens that in a set of objects that can be composed together in some way, there are objects that are for some reason simpler than others.



        In this circumstances it can be convenient to be able to find how simpler objects combine to give an object.



        Once found, one can say that the object is decomposed over the simpler objects, but no insight into how simpler objects compose is given.



        If the composition is a sum of the simpler objects each multiplied by a factor then one can use the espression "the object is expanded by the simpler objects".
        (See e.g. vector expansion, Taylor or Fourier expansion)



        If the composition is a product of the simpler objects each raised to some power then one can use the espression "the object is factored into the simpler objects".
        (See e.g. integer factorization or polynomial factorization)



        Cofactor = complement of factor



        $C_11$ is the complement of the factor $a_11$. It is the complement because it is computed by complementation of $a_11$ w.r.t. the matrix, that is, by eliminating by the matrix the row and the column containing $a_11$.






        share|cite|improve this answer













        Frequently it happens that in a set of objects that can be composed together in some way, there are objects that are for some reason simpler than others.



        In this circumstances it can be convenient to be able to find how simpler objects combine to give an object.



        Once found, one can say that the object is decomposed over the simpler objects, but no insight into how simpler objects compose is given.



        If the composition is a sum of the simpler objects each multiplied by a factor then one can use the espression "the object is expanded by the simpler objects".
        (See e.g. vector expansion, Taylor or Fourier expansion)



        If the composition is a product of the simpler objects each raised to some power then one can use the espression "the object is factored into the simpler objects".
        (See e.g. integer factorization or polynomial factorization)



        Cofactor = complement of factor



        $C_11$ is the complement of the factor $a_11$. It is the complement because it is computed by complementation of $a_11$ w.r.t. the matrix, that is, by eliminating by the matrix the row and the column containing $a_11$.







        share|cite|improve this answer













        share|cite|improve this answer



        share|cite|improve this answer











        answered Aug 6 at 19:28









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