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Let's learn the various properties of multiplication of matrix as follows:

1. Commutative property

2. Associative property

3. Distributive property

4. Multiplication of a matrix by a unit matrix

Let's dive into each property individually with an example.
Commutative property of matrix multiplication:
In general, matrix multiplication is not commutative.
If $$A$$ is of order $$m×n$$ and $$B$$ of the order $$n×p$$ then $$AB$$ is defined but $$BA$$ is not defined. Even if $$AB$$ and $$BA$$ are both defined, they don't need to be equal.
Therefore, in general $$AB ≠ BA$$.
Associative property:
Matrix multiplication always satisfies the associative property.

That is $$(AB)C = A(BC)$$.
Example:
If $$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, B = \begin{bmatrix} 5 & 6\\ 7 & 8 \end{bmatrix}, C = \begin{bmatrix} 9 & 10\\ 11 & 12 \end{bmatrix}$$ then verify that $$(AB)C = A (BC)$$.

Solution:

First, we find the sum of $$AB$$ matrices, then multiply its result with the $$C) matrix. \((AB) C= \left ( \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 5 & 6\\ 7 & 8 \end{bmatrix} \right )\begin{bmatrix} 9 & 10\\ 11 & 12 \end{bmatrix}$$

$$=\begin{bmatrix} (1×5) + (2×7) & (1×6) + (2×8)\\ (3×5) + (4×7) & (3×6) + (4×8) \end{bmatrix} \begin{bmatrix} 9 & 10\\ 11 & 12 \end{bmatrix}$$

$$=\begin{bmatrix} 5 + 14 & 6 + 18\\ 15+28 & 18+32 \end{bmatrix} \begin{bmatrix} 9 & 10\\ 11 & 12 \end{bmatrix}$$

$$=\begin{bmatrix} 19 & 24\\ 43 & 50 \end{bmatrix} \begin{bmatrix} 9 & 10\\ 11 & 12 \end{bmatrix}$$

$$=\begin{bmatrix} (19 × 9) + (24×11)& (19 × 10) + (24×12)\\ (43 × 9) + (50×11)& (43 × 10) + (50×12) \end{bmatrix}$$

$$= \begin{bmatrix} 171+264 & 190 + 288\\ 387 + 550 & 430 + 600 \end{bmatrix}=\begin{bmatrix} 435 & 478\\ 937 & 1030 \end{bmatrix}$$……….(1)

Similarly, let's find $$A(BC)$$.

$$(AB) C= \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \left (\begin{bmatrix} 5 & 6\\ 7 & 8 \end{bmatrix} \begin{bmatrix} 9 & 10\\ 11 & 12 \end{bmatrix}\right )$$

$$A( B C) =\begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix} \begin{bmatrix} (5×9)+ (6 × 11) & (5×10)+ (6 × 12)\\ (7×9)+ (8 × 11) & (7×10)+ (8 × 12) \end{bmatrix}$$

$$=\begin{bmatrix} 1& 2\\ 3& 4 \end{bmatrix} \begin{bmatrix} 111 & 122\\ 151 & 166 \end{bmatrix}$$

$$= \begin{bmatrix} (1×111)+ (2× 151) & (1×122)+ (2 × 166)\\ (3×111)+ (4× 151) & (3×122)+ (4× 166) \end{bmatrix}=\begin{bmatrix} 413 & 454\\ 937 & 1030 \end{bmatrix}$$……….(2)

From $$(1), (2)$$ we can see that $$(AB)C) = A(BC)$$.

Thus, the given matrix satisfies the associative property of matrices.