Question

In this exercise we show that matrix application is associative. Suppose that A is an m x pmatrix, B is a p x k matrix, Cis a k x n matrix. Show that A (BC) = (AB)C .

Short answer

A (BC) = (AB)C

Step-by-step solution

Step 1

Given : A is a m x p matrix, B is a p x k matrices and C is a k x n matrix.

To prove : A (BC) = (AB) C

PROOF

$\begin{array}{r}\left[A\right(BC){]}_{ij}=\sum _{a=1}^p \left(\left[A{]}_{ia}\right[BC{]}_{aj}\right)\\ =\sum _{a=1}^p \sum _{j=1}^k \left([A{]}_{iz}\left(\left[B{]}_{aj}\right[C{]}_{bj}\right)\right)\end{array}$

Associative property of the multiplication of real numbers

Step 2

The associative property states that the changing the group of adding or multiplying real numbers does not change the result.

$\begin{array}{r}=\sum _{a=1}^p \sum _{b=1}^k \left([A{]}_{iz}\left(\left[B{]}_{a\dot{j}}\right[C{]}_{ij}\right)\right)\\ =\sum _{b=1}^k \sum _{a=1}^p \left(\left(\left[A{]}_{in}\right[B{]}_{a\dot{b}}\right)[C{]}_{bj}\right)\\ =\sum _{b=1}^k \sum _{a=1}^p \left(\left[AB{]}_{\dot{\widehat{p}}}\right[C{]}_{\dot{b}j}\right)\\ =\left[\right(AB)C{]}_{ij}\end{array}$

This then implies that every element of the matrix A(BC) is identical to the corresponding element of the matrix (AB)C ,which implies that the two matrices are equal.

A (BC) = (AB)C