# Associative Properties of Matrices:

The Associative Property of Addition for Matrices states:

Let $A$, $B$ and $C$ be $m×n$ matrices.  Then, $\left(A+B\right)+C=A+\left(B+C\right)$.

Example 1:

$A=\left[\begin{array}{rrr}3& 2& 4\\ -1& 0& -5\end{array}\right],B=\left[\begin{array}{rrr}-2& 3& -1\\ 4& 2& 0\end{array}\right],C=\left[\begin{array}{rrr}8& -1& 5\\ 6& 1& 2\end{array}\right]$

Find $\left(A+B\right)+C$ and $A+\left(B+C\right)$

Find $\left(A+B\right)+C$:

$\begin{array}{l}\left(\left[\begin{array}{rrr}3& 2& 4\\ -1& 0& -5\end{array}\right]+\left[\begin{array}{rrr}-2& 3& -1\\ 4& 2& 0\end{array}\right]\right)+\left[\begin{array}{rrr}8& -1& 5\\ 6& 1& 2\end{array}\right]\\ =\left[\begin{array}{rrr}1& 5& 3\\ 3& 2& -5\end{array}\right]+\left[\begin{array}{rrr}8& -1& 5\\ 6& 1& 2\end{array}\right]\\ =\left[\begin{array}{rrr}9& 4& 8\\ 9& 3& -3\end{array}\right]\end{array}$

Find $A+\left(B+C\right)$:

$\begin{array}{l}\left[\begin{array}{rrr}3& 2& 4\\ -1& 0& -5\end{array}\right]+\left(\left[\begin{array}{rrr}-2& 3& -1\\ 4& 2& 0\end{array}\right]+\left[\begin{array}{rrr}8& -1& 5\\ 6& 1& 2\end{array}\right]\right)\\ =\left[\begin{array}{rrr}3& 2& 4\\ -1& 0& -5\end{array}\right]+\left[\begin{array}{rrr}6& 2& 4\\ 10& 3& 2\end{array}\right]\\ =\left[\begin{array}{rrr}9& 4& 8\\ 9& 3& -3\end{array}\right]\end{array}$

The Associative Property of Multiplication of Matrices states:

Let $A$, $B$ and $C$ be $n×n$ matrices.  Then, $\left(AB\right)C=A\left(BC\right)$

Example 2:

$A=\left[\begin{array}{rr}3& 2\\ -1& 0\end{array}\right],B=\left[\begin{array}{rr}-2& 3\\ 4& 2\end{array}\right],C=\left[\begin{array}{rr}-1& 5\\ 1& 2\end{array}\right]$

Find $\left(AB\right)C$ and $A\left(BC\right)$.

Find $\left(AB\right)C$:                                                      Find $A\left(BC\right)$:

$\begin{array}{rrrrr}\begin{array}{l}\left(\left[\begin{array}{rr}3& 2\\ -1& 0\end{array}\right]\left[\begin{array}{rr}-2& 3\\ 4& 2\end{array}\right]\right)\left[\begin{array}{rr}-1& 5\\ 1& 2\end{array}\right]\\ =\left[\begin{array}{rr}2& 13\\ 2& -3\end{array}\right]\left[\begin{array}{rr}-1& 5\\ 1& 2\end{array}\right]\\ =\left[\begin{array}{rr}11& 36\\ -5& 4\end{array}\right]\end{array}& & & & \begin{array}{l}\left[\begin{array}{rr}3& 2\\ -1& 0\end{array}\right]\left(\left[\begin{array}{rr}-2& 3\\ 4& 2\end{array}\right]\left[\begin{array}{rr}-1& 5\\ 1& 2\end{array}\right]\right)\\ =\left[\begin{array}{rr}3& 2\\ -1& 0\end{array}\right]\left[\begin{array}{rr}5& -4\\ -2& 24\end{array}\right]\\ =\left[\begin{array}{rr}11& 36\\ -5& 4\end{array}\right]\end{array}\end{array}$