A mathematical approach of adding a matrix to another matrix is called addition of matrices.

Matrices are often involved in addition. So, it can be done by adding elements in one matrix with the corresponding elements of another matrix. Remember, it is not possible to perform the summation of matrices if they are of the different order.

$A = $ $\begin{bmatrix}
1 & 4 \\

5 & 7 \\

\end{bmatrix}$ and $B = $ $\begin{bmatrix}
3 & 2 \\

6 & 3 \\

\end{bmatrix}$ are two matrices.

The two matrices are second order matrices and each matrix contains four elements. Now, add both matrices and the addition of them is written as $A+B$.

$\implies A+B$ $\,=\,$ $\begin{bmatrix}
1 & 4 \\

5 & 7 \\

\end{bmatrix}$ $+$ $\begin{bmatrix}
3 & 2 \\

6 & 3 \\

\end{bmatrix}$

Add each element of one matrix to corresponding element of the other matrix.

$\implies A+B$ $\,=\,$ $\begin{bmatrix}
1+3 & 4+2 \\

5+6 & 7+3 \\

\end{bmatrix}$

$\,\,\, \therefore \,\,\,\,\,\, A+B$ $\,=\,$ $\begin{bmatrix}
4 & 6 \\

11 & 10 \\

\end{bmatrix}$

The addition of matrices is successful due to the addition of same order matrices. In this way, two or more matrices can be added mathematically to get the sum of them.

Now, try to add matrices which belong to different orders for understanding the impossibility of adding matrices of different order.

$C = $ $\begin{bmatrix}
6 & 1 & 3 \\

2 & 2 & 5 \\

\end{bmatrix}$ and $D = $ $\begin{bmatrix}
9 & 2 \\

4 & 7 \\

\end{bmatrix}$ are two matrices of different order.

The matrix $C$ is an order of $2 \times 3$ and $D$ is a matrix of order $2$. The matrix $C$ contains six elements and the matrix $D$ has four elements.

- $6$ and $9$ are elements belong to first row, first column of matrices $C$ and $D$ respectively. Add the first row, first column elements as a first step of adding matrices.
- $1$ and $2$ are elements belong to first row, second column of matrices $C$ and $D$ respectively. Add the first row, second column elements as a second step of adding matrices.
- $3$ is an element of first row, third column of matrix $C$ but there is no element of matrix $D$ in this position. So, the matrix $C$ and $D$ cannot be added further. So, the addition of matrices $C$ and $D$ is failed.

The addition of matrices of different order is failed in three steps for this example. In this way, the matrices are failed to add each other when their order are different. It proves that the matrices are added only when they are of the same order.

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