Question

Use index notation as in (9.9) to prove the second part of the associative law for matrix multiplication: (AB)C = ABC

Short answer

The second part of associative law for matrix is proved by showing that ${\left[\left(\mathrm{AB}\right)\mathrm{C}\right]}_{\mathrm{ij}}={\left[\mathrm{ABC}\right]}_{\mathrm{ij}}$.

Step-by-step solution

Multiplication of Matrices

The element in row i and column j of the product matrix AB is equal to i row of A times column j of B . In index notation, ${\left(\mathrm{AB}\right)}_{\mathbf{ij}}\mathbf{=}\underset{\mathbf{k}}{\mathbf{\sum }{\mathbf{A}}_{\mathbf{ik}}{\mathbf{B}}_{\mathbf{kj}}}$

Prove second part of associative law

The index notation is to be used to prove the second part of associative law of matrix multiplication.

Take as the product and as matrix and use the index notation for matrix multiplication to state that${\left(\mathrm{MN}\right)}_{\mathrm{ij}}=\sum _{\mathrm{k}}{\mathrm{M}}_{\mathrm{ik}}{\mathrm{N}}_{\mathrm{kj}}$.

Substitute M = AB and N = C .

${\left[\left(\mathrm{AB}\right)\mathrm{C}\right]}_{\mathrm{ij}}=\sum _{\mathrm{k}}{\left(\mathrm{AB}\right)}_{\mathrm{ik}}{\mathrm{C}}_{\mathrm{kj}}$

Again, use the index notation for the multiplication of and.

${\left(\mathrm{AB}\right)}_{\mathrm{ik}}={\mathrm{A}}_{\mathrm{il}}{\mathrm{B}}_{\mathrm{ik}}$

Therefore,$\left[\left(\mathrm{AB}\right)\mathrm{C}\right]$can be simplified as follows:

$$\begin{gathered} {\left[\left(\mathrm{AB}\right)\mathrm{C}\right]}_{\mathrm{ij}}=\sum _{\mathrm{k}}{\left(\mathrm{AB}\right)}_{\mathrm{ik}}{\mathrm{C}}_{\mathrm{kj}} \\ =\sum _{\mathrm{k}}\left[\sum _{\mathrm{l}}{\mathrm{A}}_{\mathrm{il}}{\mathrm{B}}_{\mathrm{ik}}\right]{\mathrm{C}}_{\mathrm{kj}} \\ =\sum _{\mathrm{k}}\sum _{\mathrm{l}}{\mathrm{A}}_{\mathrm{il}}{\mathrm{B}}_{\mathrm{ik}}{\mathrm{C}}_{\mathrm{kj}} \\ ={\left(\mathrm{ABC}\right)}_{\mathrm{ij}} \end{gathered}$$

Hence, the second part of associative law for matrix,$\left(\mathrm{AB}\right)\mathrm{C}=\mathrm{ABC}$ , has been proved.