### Theory:

Matr4ices are also follows the certain properties such as whole numbers and integers. Generally, there are four properties of matrix in terms of addition and scalar multiplication as follows:

1. Commutative property

2. Associative property

3. Scalar identity for unit matrix

4. Distributive property

Let's dive into each property individually with an example.

Let $$A$$, $$B$$, $$C$$ be $$m×n$$ matrices and $$p$$ and $$q$$ be two non-zero scalars (numbers). Then we have the following properties.
Changing the order of the matrices does not change the result of the matrices.

$$A + B = B + A$$
Example:
Consider the matrices $$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, B = \begin{bmatrix} 5 & 6\\ 7 & 8 \end{bmatrix}$$ then verify that $$A + B = B + A$$

Solution:

First we find the sum of $$A + B$$ matrices.

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

Similarly, let find $$B + A$$.

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

From $$(1), (2)$$ we can see that $$A + B = B + A$$. Thus, the given matrices are satisfy the Commutative property of matrices.
Associative property:
1. Associative property of matrix addition:

It says that rearranging parenthesis in the matrix expression will not change the result of the matrices.

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

Solution:

First we find the sum of $$A + B$$ matrices then add it's result with the $$C) matrix. \((A + B) + 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}$$

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

$$=\begin{bmatrix} 6 & 8\\ 10 & 12 \end{bmatrix}+ \begin{bmatrix} 9 & 10\\ 11 & 12 \end{bmatrix}$$

$$=\begin{bmatrix} 6+9 & 8+10\\ 10+11 & 12+12 \end{bmatrix}= \begin{bmatrix} 15 & 18\\ 21 & 24 \end{bmatrix}$$……….(1)

Similarly, let's find $$A + (B + C)$$.

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

$$=\begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix}+ \begin{bmatrix} 14 & 16\\ 18 & 20 \end{bmatrix}$$

$$= \begin{bmatrix} 1+14 & 2+16\\ 18+3 & 4+20 \end{bmatrix}=\begin{bmatrix} 15 & 18\\ 21 & 24 \end{bmatrix}$$……….(2)

From $$(1), (2)$$ we can see that $$(A + B) + C = A + (B+C)$$. Thus, the given matrices satisfy the Associative property of matrices.
2. Associative property of scalar multiplication - $$(pq)A = p(Aq)$$
Example:
Verify the associative property if $$A = \begin{bmatrix} 2 & 4\\ 6 & 8 \end{bmatrix}$$ and $$p = 4$$ and $$q = 6$$.

Solution:

We know the associative property of scalar multiplication is  $$(pq)A = p(Aq)$$. Here, $$p = 4$$ and $$q = 6$$.

So,  $$(4 × 6) A = (4 × 6) \begin{bmatrix} 2 & 4\\ 6 & 8 \end{bmatrix}$$

$$= 24\begin{bmatrix} 2 & 4\\ 6 & 8 \end{bmatrix} = \begin{bmatrix} 48 & 96\\ 144 & 192 \end{bmatrix}$$……….(1)

Similarly, let's find $$4( A × 6) = 4 (6 ×\begin{bmatrix} 2 & 4\\ 6 & 8 \end{bmatrix})$$

$$= 4 × \begin{bmatrix} 12 & 24\\ 36 & 48 \end{bmatrix} = \begin{bmatrix} 48 & 96\\ 144 & 192 \end{bmatrix}$$……..(2)

From the equations $$(1)$$ and $$(2)$$, we can see that $$(pq)A = p(Aq)$$.

Hence, the given matrix follows the associative property of scalar multiplication.