Question

Show that matrix addition is associative; that is, show

that if A, B, and C are all m × n matrices, then

A + (B + C) = (A + B) + C.

Short answer

A + (B + C) = (A + B) + C

Step-by-step solution

Step 1:

A,B and C are matrices

A + (B + C) = (A + B) + C

We can rewrite the matrix A as$\left[a_{ij}\right]$ where$a_{ij}$ represent the element in the ith row and jth column of A

The addition of two matrices add the corresponding elements In the matrices( thus the two elements that are In the same row and column in the two matrices

Step 2:

$\begin{array}{r}A+(B+C)=\left[a_{ij}\right]+\left[b_{ij}\right]+\left[c_{ij}\right]\\ =\left[a_{ij}\right]+\left[b_{ij}+c_{ij}\right]\\ =\left[a_{ij}+\left(b_{ij}+c_{ij}\right)\right]\end{array}$

Use the associative property of addition of real numbers

$\begin{array}{r}=\left[\left(a_{ij}+b_{ij}\right)\right]+\left[c_{ij}\right]\\ =\left[a_{ij}+b_{ij}\right]+\left[c_{ij}\right]\\ =\left[a_{ij}\right]+\left[b_{ij}\right]+\left[c_{ij}\right]\\ =(A+B)+C\end{array}$