Broadcasting Error in NumPy Matrix Multiplication

When performing matrix multiplication between two NumPy arrays, it is crucial to ensure that their shapes are compatible. However, the incorrect use of the multiplication operator (*) can lead to the "ValueError: operands could not be broadcast together with shapes" error.

Consider two arrays: X with shape (m, n) and y with shape (n, 1). The intended operation is matrix multiplication, which should result in an (m, 1) vector. However, the multiplication operator in NumPy is used for element-wise operations, which requires broadcasting to align the dimensions.

In the given example, the broadcasting rules are violated because the first dimension of X (97) conflicts with the corresponding dimension of y (2). Therefore, the multiplication operation fails and triggers the error.

To perform matrix multiplication, you should use the dot() method. This method is specifically designed for matrix operations and ensures proper alignment of the dimensions. Using dot() instead of * resolves the broadcasting issue and produces the expected (m, 1) vector.

Additionally, note that using the matrix type in NumPy (numpy.matrix) can complicate operations. It is generally recommended to use numpy.ndarray for arrays instead.