# In matrix multiplication, does ABC=ACB?

Mar 30, 2018

No, at least not in general

#### Explanation:

Matrix multiplication is associative but not generally commutative.

Let:

$A = \left(\begin{matrix}1 & 0 \\ 0 & 1\end{matrix}\right)$

$B = \left(\begin{matrix}1 & 1 \\ 0 & 1\end{matrix}\right)$

$C = \left(\begin{matrix}1 & 0 \\ 1 & 1\end{matrix}\right)$

Then:

$A B C = B C = \left(\begin{matrix}1 & 1 \\ 0 & 1\end{matrix}\right) \left(\begin{matrix}1 & 0 \\ 1 & 1\end{matrix}\right) = \left(\begin{matrix}2 & 1 \\ 1 & 1\end{matrix}\right)$

$A C B = C B = \left(\begin{matrix}1 & 0 \\ 1 & 1\end{matrix}\right) \left(\begin{matrix}1 & 1 \\ 0 & 1\end{matrix}\right) = \left(\begin{matrix}1 & 1 \\ 1 & 2\end{matrix}\right)$