definition of \((j, l)\)-cofactor of square matrix
Topics
About: matrices space
The table of contents of this article
Starting Context
- The reader knows a definition of \((j, l)\)-minor of matrix.
- The reader knows a definition of determinant of square matrix over ring.
Target Context
- The reader will have a definition of \((j, l)\)-cofactor of square matrix.
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:
\( R\): \(\in \{\text{ the rings }\}\)
\( n\): \(\in \mathbb{N} \setminus \{0, 1\}\)
\( M\): \(\in \{\text{ the } n \times n R \text{ matrices }\}\)
\( j\): \(\in \{1, ..., n\}\)
\( l\): \(\in \{1, ..., n\}\)
\( M^{j, l}\): \(= \text{ the } (j, l) \text{ -minor of } M\)
\(*M_{j, l}\): \(= (-1)^{j + l} det M^{j, l}\)
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Conditions:
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2: Note
We denote any element of \(M\) always like \(M^j_l\), because \(M^{j, l}\) and \(M_{j, l}\) will be understood as the \((j, l)\)-minor and the \((j, l)\)-cofactor.
While the \((j, l)\)-minor of a non-square matrix is possible, the \((j, l)\)-cofactor of any non-square matrix is not possible, because \(det M^{j, l}\) would not be valid.
