definition of inverse of square matrix over ring
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 definition of inverse of square matrix over ring.
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:
\( R\): \(\in \{\text{ the rings }\}\)
\( n\): \(\in \mathbb{N} \setminus \{0\}\)
\( M\): \(\in \{\text{ the } n \times n R \text{ matrices }\}\)
\(*M^{-1}\): \(\in \{\text{ the } n \times n R \text{ matrices }\}\)
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Conditions:
\(M^{-1} M = I \land M M^{-1} = I\)
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2: Note
This definition is not claiming that there is an \(M^{-1}\) for any \(M\).
When there is an \(M^{-1}\), it is the unique inverse, because supposing there is another inverse, \(N\), \(M N = I\), but \(M^{-1} M N = M^{-1} I\), and the left hand side is \(I N = N\) and the right hand side is \(M^{-1}\), so, \(N = M^{-1}\).
