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## Matrix algebra

• By convention, we define the product of a row vector and a column vector as a scalar:

This is the foundation of the correspondence between linear equations and matrices. It is also why matrix multiplication is defined as it is: a matrix is considered as a collection of row vectors.

• The above multiplication rule, together with the obvious addition rules, turn the set of n by n matrices into a ring. We observe that multiplication is not commutative, and that matrices do not necessarily have multiplicative inverses, so this ring is not a field and is not a commutative ring.
• Matrix multiplication can usefully be viewed in several ways:
1. As the row-vector times column-vector product just noted.
2. A B can be thought of as the result of the matrix A applied to the columns of B: which is .
3. As the so-called outer product given by column-vectors times row-vectors:

Robert Corless
Wed Jan 31 11:33:59 EST 1996