Matrix Multiplication is Associative

Proved by: The ith Column of AB is Abi, The ith Row of AB is the Product of The ith Row of A and Matrix B
References: Not Applicable
Justifications: Not Applicable

Specializations: Not Applicable
Generalizations: Not Applicable

Matrix Multiplication is Associative

If is an Matrix, is an matrix, and a matrix, so that and are both defined, then they are equal:

Proof.
Since The ith Column of AB is Abi and The ith Row of AB is the Product of The ith Row of A and Matrix B, the the entry of both and depend on only the th row of and the th column of (but on all the entries of ). Without loss of generality we can assume that is a line matrix and is a column matrix (), so that both and are numbers. Now apply the associativity of multiplication of numbers: