Matrix Multiplication is Associative

Showing \( \left( A B \right) C = A \left( B C \right) \) is relatively a simple matter using the super duper summation convention.
Firstly the i^th row, j^th column of a matrix \( A \) is designated by \( A_{ij} \), \( A^{i}_{j} \) or \( A^{ij} \). Matrix multiplication is then defined as $$ \left[ XY \right]_{ij} = X_{ik}Y_{kj} $$ where square brackets have been placed around \( X Y \) to designate the \( i j \) element of the matrix \( X \cdot Y \). \begin{aligned} \left[ \left( A B \right) C \right]_{ij} & = \left( A B \right)_{ik} C_{kj} \\ & = A_{il} B_{lk} C_{kj} \\ & = A_{il} \left( B_{lk} C_{kj} \right) \\ & = A_{il} \left[ B C \right]_{lj} \\ & = \left[ A \left( B C \right) \right]_{ij} \\ \therefore \left( A B \right) C & = A \left( B C \right) \end{aligned}