description/proof of that for ring, multiplications of matrices over ring are associative
Topics
About: ring
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 ring, the multiplications of any matrices over the ring are associative.
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:
\(R\): \(\in \{\text{ the rings }\}\)
\(M_1\): \(\in \{\text{ the } n_1 \times n_2 \text{ matrices over } R\}\)
\(M_2\): \(\in \{\text{ the } n_2 \times n_3 \text{ matrices over } R\}\)
\(M_3\): \(\in \{\text{ the } n_3 \times n_4 \text{ matrices over } R\}\)
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Statements: \((M_1 M_2) M_3 = M_1 (M_2 M_3)\)
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2: Note
So, being associative is not only for \(R\) real or complex or even any field but always for \(R\) any ring.
3: Proof
Whole Strategy: Step 1: see that \(((M_1 M_2) M_3)^j_n = (M_1 (M_2 M_3))^j_n\).
Step 1:
\(((M_1 M_2) M_3)^j_n = \sum_m ((\sum_l {M_1}^j_l {M_2}^l_m) {M_3}^m_n) = \sum_m (\sum_l ({M_1}^j_l {M_2}^l_m {M_3}^m_n))\), by the multiplication distributability with respect to addition of \(R\).
\((M_1 (M_2 M_3))^j_n = \sum_l ({M_1}^j_l \sum_m ({M_2}^l_m {M_3}^m_n)) = \sum_l (\sum_m ({M_1}^j_l {M_2}^l_m {M_3}^m_n))\), by the multiplication distributability with respect to addition of \(R\).
\(\sum_m (\sum_l ({M_1}^j_l {M_2}^l_m {M_3}^m_n)) = \sum_l (\sum_m ({M_1}^j_l {M_2}^l_m {M_3}^m_n))\), because each term of the left hand side appears only once as a term in the right hand side and each term of the right hand side appears only once as a term in the left hand side, while addition in \(R\) is commutative.
So, \(((M_1 M_2) M_3)^j_n = (M_1 (M_2 M_3))^j_n\).
So, \((M_1 M_2) M_3 = M_1 (M_2 M_3)\).
