Explain with example the properties of matrix multiplication. 

Matrix multiplication is a fundamental operation in linear algebra that involves multiplying two matrices to produce a new matrix. The properties of matrix multiplication are crucial in various fields, including computer graphics, engineering, physics, and economics. In this explanation, we will explore the key properties of matrix multiplication along with illustrative examples.

Let's consider two matrices, A and B, and assume they are compatible for multiplication. Matrix A has dimensions (m x n), meaning it has m rows and n columns, while matrix B has dimensions (n x p), with n rows and p columns. The resulting matrix C, after multiplication, will have dimensions (m x p).

Now, let's delve into the properties of matrix multiplication:

1. Associativity: Matrix multiplication is associative, which means the order of multiplication does not affect the final result. In other words, for matrices A, B, and C of appropriate dimensions, (AB)C = A(BC).

Example: Consider three matrices, A (2x3), B (3x2), and C (2x4):

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