Pairwise Matrix Product Computation

Matrix multiplication is one of the most fundamental operations in linear algebra and serves as the backbone for countless algorithms in machine learning, computer graphics, physics simulations, and data science. When two matrices are multiplied together, the result encodes a composition of linear transformations, enabling complex multi-step transformations to be expressed as a single matrix operation.

Given two matrices A of dimensions n × m and B of dimensions p × q, the product C = A · B is defined only when the number of columns in A equals the number of rows in B (i.e., m = p). When this condition is satisfied, the resulting matrix C has dimensions n × q.

Each element C[i][j] of the resulting matrix is computed as the dot product of the i-th row of matrix A and the j-th column of matrix B:

$$C_{ij} = \sum_{k=1}^{m} A_{ik} \cdot B_{kj}$$

Dimensional Compatibility Rule:

• Matrix A: n × m (n rows, m columns)
• Matrix B: p × q (p rows, q columns)
• For multiplication to be valid: m must equal p
• Resulting matrix C: n × q (n rows, q columns)

Important Properties:

• Matrix multiplication is associative: (A · B) · C = A · (B · C)
• Matrix multiplication is NOT commutative: A · B ≠ B · A in general
• The identity matrix I acts as a neutral element: A · I = I · A = A

Your Task: Write a Python function that computes the product of two matrices. The function should:

• Return a 2D list representing the resulting matrix if the dimensions are compatible
• Return -1 if the matrices cannot be multiplied (i.e., the number of columns in the first matrix does not equal the number of rows in the second matrix)