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Matrix Multiplication Rules

The rules governing matrix multiplication are essential for applying operators to wavefunctions, ensuring the correct transformation of quantum states.
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The statement of the theorem

Let A∈Cm×k\mathbf{A} \in \mathbb{C}^{m \times k} and B∈Ck×n\mathbf{B} \in \mathbb{C}^{k \times n} be two matrices with complex entries. The product matrix C=AB∈Cm×n\mathbf{C} = \mathbf{A} \mathbf{B} \in \mathbb{C}^{m \times n} is defined by its entries cijc_{ij} as:\n\nC=AB  ⟹  cij=∑l=1kailblj\mathbf{C} = \mathbf{A} \mathbf{B} \implies c_{ij} = \sum_{l=1}^{k} a_{il} b_{lj}
Source: Wikipedia