Multiplication of real valued square matrices of same dimension is

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Multiplication of real-valued square matrices of the same dimension has several important properties:

1. **Associativity**:
   - Matrix multiplication is associative. This means that for any three $$ n \times n $$ matrices $$ A $$, $$ B $$, and $$ C $$:
     $$
     (A \cdot B) \cdot C = A \cdot (B \cdot C)
     $$
   
2. **Distributivity**:
   - Matrix multiplication is distributive over matrix addition. For any three $$ n \times n $$ matrices $$ A $$, $$ B $$, and $$ C $$:
     $$
     A \cdot (B + C) = A \cdot B + A \cdot C
     $$
     $$
     (A + B) \cdot C = A \cdot C + B \cdot C
     $$
   
3. **Non-Commutativity**:
   - In general, matrix multiplication is **not commutative**. That is, for two $$ n \times n $$ matrices $$ A $$ and $$ B $$, it is not necessarily true that:
     $$
     A \cdot B = B \cdot A
     $$
     However, there are special cases where matrices do commute, but this is not the norm.
   
4. **Closure**:
   - The set of $$ n \times n $$ real matrices is closed under matrix multiplication. This means that multiplying any two $$ n \times n $$ matrices will result in another $$ n \times n $$ matrix.

5. **Identity Element**:
   - There exists an identity matrix $$ I $$ such that for any $$ n \times n $$ matrix $$ A $$:
     $$
     I \cdot A = A \cdot I = A
     $$
   
6. **Inverses**:
   - Not all square matrices have inverses. A matrix $$ A $$ has an inverse $$ A^{-1} $$ if and only if it is non-singular (i.e., its determinant is non-zero).

**Summary**:
Multiplication of real-valued square matrices of the same dimension is **associative** and **distributive** over addition, **closed** within the set of such matrices, and possesses an **identity element**. However, it is **not generally commutative**, and not all matrices have multiplicative inverses.

**Example**:
Consider the following two $$ 2 \times 2 $$ matrices:
$$
A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 5 & 6 \ 7 & 8 \end{pmatrix}
$$

Compute $$ A \cdot B $$:
$$
A \cdot B = \begin{pmatrix} (1)(5) + (2)(7) & (1)(6) + (2)(8) \ (3)(5) + (4)(7) & (3)(6) + (4)(8) \end{pmatrix} = \begin{pmatrix} 19 & 22 \ 43 & 50 \end{pmatrix}
$$

Compute $$ B \cdot A $$:
$$
B \cdot A = \begin{pmatrix} (5)(1) + (6)(3) & (5)(2) + (6)(4) \ (7)(1) + (8)(3) & (7)(2) + (8)(4) \end{pmatrix} = \begin{pmatrix} 23 & 34 \ 31 & 46 \end{pmatrix}
$$

As shown, $$ A \cdot B \neq B \cdot A $$, illustrating that matrix multiplication is not commutative in general.
Multiplying real-valued square matrices of the same size is associative and distributive over addition. It's not commutative in general, meaning the order of multiplication matters. There's an identity matrix that acts as a neutral element, and not all matrices have inverses.

Multiplication of real valued square matrices of same dimension is

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