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.
Commutative property of matrix addition:
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.