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In this section, we shall discuss the addition and subtraction of matrices, multiplication of a matrix by a scalar and multiplication of matrices.
Addition and subtraction of matrices:
• Two matrices can be added or subtracted if they have the same order.
• To add or subtract two matrices, add or subtract the corresponding elements.
Example:

$$\begin{bmatrix} a & b & c\\ d & e & f \end{bmatrix} + \begin{bmatrix} g & h & i\\ j & k & l \end{bmatrix} = \begin{bmatrix} a+g & b+h & c+i\\ d+j & e+k & f+l \end{bmatrix}$$

Subtraction in matrices:

$$\begin{bmatrix} a & b\\ c & d \end{bmatrix} - \begin{bmatrix} e & f\\ g & h \end{bmatrix} = \begin{bmatrix} a-e & b-f\\ c-g & d-h \end{bmatrix}$$
If $$A = (a_{ij})$$,  $$B = (b_{ij})$$, $$i = 1, 2,...m$$, and $$j = 1, 2, ….n$$

Condition of addition of matrices:
The order of the matrices should be equal when adding the matrices.
Example:
Let's look at the below two matrices.

$$A = \begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix}, B =\begin{bmatrix} 1 & 2\\ 3 & 4\\ 5 & 6 \end{bmatrix}$$ what will be the sum of $$A$$ and $$B$$?

Solution:

Observing the above two matrices we can see that, the matrix $$A$$ have $$3$$-columns and $$3$$-rows but the matrix $$B$$ have $$3$$-columns and $$2$$-rows only.

$$A = (a_{ij}) = 3 × 3$$ matrices and $$B = (a_{ij}) = 3 × 2$$.

When the orders of the two matrices are not equal, then we cannot perform the addition of the same matrices.

Therefore, it is impossible to add $$A$$ and $$B$$ since they have different orders.