Chapter 2.1, Problem 3P

Explanation of Solution

Proving matrix multiplication is associative:

Suppose that there are three matrices A, B, and C such that matrix A has elements $a_{ij}$, matrix B has elements $b_{ij}$, and matrix C has elements $c_{ij}$.

The product BC is defined because the columns of B and rows of C are same.

Suppose D=BC. Then, the element of D is of the form given below:

$d_{ik}={\displaystyle \sum _j{b}_{ij}{c}_{jk}}$

The product AD is defined because the columns of A and rows of D are same.

Suppose E=AD. Then, the element of E is of the form given below:

$e_{lk}={\displaystyle \sum _k{a}_{lk}}\left({\displaystyle \sum _j{b}_{ij}{c}_{jk}}\right)\mathrm{..........}(1)$

Therefore, the matrix E=A(BC) have the elements $e_{lk}$