# Is matrix multiplication associative

One of the properties of matrix multiplication is associative property of multiplication.

Consider 3 matrices A, B and C, if and only if the total number of columns of A = total number of rows of B, and the total number of columns of B = total number of rows of C.

The property states that

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

For instance, if matrix A and B are multiplied, then the resultant matrix is multiplied with C, or matrix B and C are multiplied and then the resultant matrix is multiplied with matrix A, both the results of multiplication should be the same.

$$\\A=\begin{bmatrix} 3 &2 \\ -1&0 \end{bmatrix};B=\begin{bmatrix} -2 &3 \\ 4&2 \end{bmatrix};C=\begin{bmatrix} -1 &5 \\ 1&2 \end{bmatrix}\\ (A \times B) \times C =(\begin{bmatrix} 3 &2 \\ -1&0 \end{bmatrix}\times \begin{bmatrix} -2 &3 \\ 4&2 \end{bmatrix})\times \begin{bmatrix} -1 &5 \\ 1&2 \end{bmatrix}\\ =\begin{bmatrix} 2 &13 \\ 2&-3 \end{bmatrix}\begin{bmatrix} -1 &5 \\ 1&2 \end{bmatrix}\\ =\begin{bmatrix} 11 &36 \\ -5&4 \end{bmatrix}$$ $$\\ A \times (B \times C) =\begin{bmatrix} 3 &2 \\ -1&0 \end{bmatrix}\times (\begin{bmatrix} -2 &3 \\ 4&2 \end{bmatrix} \times \begin{bmatrix} -1 &5 \\ 1&2 \end{bmatrix})\\ =\begin{bmatrix} 3 &2 \\ -1&0 \end{bmatrix}\begin{bmatrix} 5 &-4 \\ -2&24 \end{bmatrix}\\ =\begin{bmatrix} 11 &36 \\ -5&4 \end{bmatrix}$$

Hence (A . B) . C = A . (B . C).

Therefore, matrix multiplication is associative.

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