Matrices originated in the study of linear equations but have now acquired an independent status and constitute one of the most important systems in algebra. They are a rich source of examples of algebraic structures, which we shall study later.

As usual, for any positive integer *n*, the set {1,…,*n*} will be denoted by **n.** The cartesian product m × n is therefore the set

**Definition.** *Let F be a field and m and n positive integers. A mapping*

*is called an* m **×** n matrix *over F*.

Let *A* be an *m* **×** *n* matrix ...

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