description/proof of for commutative ring, some expansions of determinant of product of rectangular matrices
Topics
About: matrices space
The table of contents of this article
Starting Context
- The reader knows a definition of %ring name% matrices space.
Target Context
- The reader will have a description and a proof of the proposition that for any commutative ring, some expansions of the determinant of the product of any rectangular matrices whose product is square hold.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
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Statements:
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2: Note
The ring needs to be commutative, because "well-known properties of determinant" used in Proof requires it.
The expansions claim that when
For example, when
As any square matrix is a rectangular matrix,
In that case, also
For the
3: Proof
Whole Strategy: Step 1: see the components of
Step 1:
Step 2:
The 1st row of
By a well-known property of determinant,
By a well-known property of determinant,
For each of the new determinants, the 2nd row can be written as
By a well-known property of determinant,
By a well-known property of determinant,
So,
And so on, after all,
Step 3:
By the result of Step 2,