You've learned how to define matrices and represent them in Python using NumPy. Now, it's time to manipulate these structures. This chapter focuses on the fundamental arithmetic and structural operations applied to matrices, which are frequently used in data processing and formulating machine learning models.

You will learn how to perform:

• Matrix Addition and Subtraction: Combining matrices element-wise ($A + B$ and $A - B$).
• Scalar Multiplication: Scaling a matrix by a single number ($c A$).
• Matrix Transpose: Flipping a matrix over its diagonal ($A^T$).
• Matrix Multiplication: Combining two matrices according to specific rules ($A B$), a key operation in many algorithms.

We will discuss the requirements for each operation (like matching dimensions) and their properties. Naturally, we will continue using NumPy to perform these calculations efficiently in code. Mastering these operations is key to understanding how linear algebraic concepts translate into practical computations.