Math 2.0

Linear and multilinear algebra; matrix theory Proposition

Associativity of matrix multiplication

Builds on Matrix multiplication

Statement

Let RR be a ring and let m,n,p,q≥0m, n, p, q \ge 0 be integers. Let AA be an m×nm \times n matrix, BB an n×pn \times p matrix and CC a p×qp \times q matrix over RR. Then (AB)C=A(BC)(AB)C = A(BC), where the products are matrix products.

Remarks

Both sides have entry (i,l)(i, l) equal to ∑j=1n∑k=1paijbjkckl\sum_{j=1}^{n} \sum_{k=1}^{p} a_{ij} b_{jk} c_{kl}. Because of this property, the product ABCABC needs no parentheses. The property also follows from the associativity of the composition of linear maps.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

Show the path as a graph
Matrix, definition, no evidence levelMatrixdefinitionMatrix multiplication, definition, no evidence levelMatrix multiplicationdefinitionAssociativity of matrix multiplication, statement, no evidence levelAssociativity of matr…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Matrix definition
  2. Matrix multiplication definition
  3. Associativity of matrix multiplication statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Matrix_multiplication, General properties
About this page
Tag073E4XC016
URIm2://073E4XC016
Kindstatement, proposition
WritingW0 stub
MSC15A99
StatusThis copy of the site has no build data. Run m2 status 073E4XC016.
UsesMatrix multiplication

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.