Let \(\mathbf{A}\) and \(\mathbf{B}\) be two \(m \times n\) matrices. The sum of \(\mathbf{A}\) and \(\mathbf{B}\), denoted by \(\mathbf{A} + \mathbf{B}\), is the \(m \times n\) matrix defined by,

\[(\mathbf{A} + \mathbf{B})_{i,j} = \mathbf{A}_{i,j} + \mathbf{B}_{i,j}, \quad 1 \leq i \leq m, \quad 1 \leq j \leq n\]