Addition of Matrices

Two matrices \(\mathbf{A}\) and \(\mathbf{B}\) can be added up, if they have the same number of rows and same number of columns. In that case the resulting matrix \(\mathbf{C} = \mathbf{A}+\mathbf{B}\) is calculated so that the corresponding elements of these matrices are summed separately. Example of matrix addition \begin{eqnarray} \left[\begin{matrix} 1&2&3\\ 4&5&6 \end{matrix} \right] + \left[\begin{matrix} 1&-2&0\\ 3&-6&7 \end{matrix} \right] &=& \left[ \begin{matrix} 2&0&3\\ 7&-1&13 \end{matrix} \right] \end{eqnarray} The rule of summing two matrices applies to any finite number of addend. The following example shows addition of 3 vector-column matrices. \begin{eqnarray} \left(\begin{matrix}2\\ 1\\3\end{matrix}\right) + \left(\begin{matrix}1\\ 1 \\ 1\end{matrix}\right) +\left(\begin{matrix}4\\ 0\\ 1\end{matrix}\right) &=& \left(\begin{matrix}7\\ 2\\ 5\end{matrix}\right) \end{eqnarray} For matrix addition the law of commutation and association law apply. \begin{eqnarray} \mathbf{A} + \mathbf{B} &=& \mathbf{B} + \mathbf{A},\\ \mathbf{A} + (\mathbf{B} + \mathbf{C}) &=& (\mathbf{A}+\mathbf{B})+\mathbf{C}. \end{eqnarray}