Multiplication of real valued square matrices of same dimension is

The correct option is associative.

Let us understand why this is the correct answer step-by-step.

1. Understanding Matrix Multiplication
Let $A$, $B$, and $C$ be three real-valued square matrices of the same dimension, say $n\times n$. The product of square matrices of the same dimension is always well-defined and results in another square matrix of the same dimension $n\times n$.

2. Defining Associativity
An operation is said to be associative if the grouping of the operands does not affect the final result. In the context of matrix multiplication, this means that for any three square matrices $A$, $B$, and $C$ of the same size, the following equality must always hold:
$\left(AB\right)C=A\left(BC\right)$
This property allows us to compute the product of multiple matrices without worrying about how we group adjacent multiplications (as long as we preserve their relative order, since matrix multiplication is not generally commutative).

3. Mathematical Proof of Associativity
Let the elements of matrices $A$, $B$, and $C$ be denoted by $a_{ij}$, $b_{jk}$, and $c_{kl}$ respectively.
First, let us find the entry at position $(i,l)$ of the matrix $\left(AB\right)C$:
The $(i,k)$-th entry of the product matrix $X=AB$ is given by:
$x_{ik}=\sum _{j=1}^n{a}_{ij}{b}_{jk}$
Now, multiplying this resulting matrix by $C$, the $(i,l)$-th entry of $\left(AB\right)C$ is:
${\left[\right(AB\left)C\right]}_{il}=\sum _{k=1}^n{x}_{ik}{c}_{kl}=\sum _{k=1}^n\left(\sum _{j=1}^n{a}_{ij}{b}_{jk}\right)c_{kl}$
Using the distributive law of real numbers, we can distribute $c_{kl}$ into the summation:
${\left[\right(AB\left)C\right]}_{il}=\sum _{k=1}^n\sum _{j=1}^n{a}_{ij}{b}_{jk}{c}_{kl}$
Since addition of real numbers is commutative and associative, we can interchange the order of the summations:
${\left[\right(AB\left)C\right]}_{il}=\sum _{j=1}^n\sum _{k=1}^n{a}_{ij}{b}_{jk}{c}_{kl}$
We can factor out $a_{ij}$ from the inner summation over $k$ since it does not depend on $k$:
${\left[\right(AB\left)C\right]}_{il}=\sum _{j=1}^n{a}_{ij}\left(\sum _{k=1}^n{b}_{jk}{c}_{kl}\right)$
Notice that the term in the parentheses is the definition of the $(j,l)$-th entry of the product matrix $BC$. Therefore:
${\left[\right(AB\left)C\right]}_{il}={\left[A\right(BC\left)\right]}_{il}$
Since this identity holds for every coordinate $(i,l)$, the matrix products are identical. Thus, matrix multiplication is associative.