Let \(\mathbf{A}\) be an \(m \times n\) matrix and \(\mathbf{B}\) be an \(n \times p\) matrix. The product of \(\mathbf{A}\) and \(\mathbf{B}\), denoted by \(\mathbf{AB}\), is the \(m \times p\) matrix defined by,

\[[\mathbf{AB}]_{i,j} = \sum_{r=1}^{n} a_{i,r}b_{r,j}, \quad 1 \leq i \leq m, \quad 1 \leq j \leq p\nonumber\]