Two matrices A and B can be added if both the matrices are of the same order $$(m \times n)$$, which means the number of rows and columns of both the matrices should be equal. The sum of the matrices will be the addition of the corresponding elements of both matrices A and B. The order of the final added matrix will always be the same $$(m \times n)$$ as the order of matrices A and B.

Example(1): Find the sum of two matrices $$A = \begin{bmatrix} 3 & 4 \\ 5 & 6 \\ \end{bmatrix}$$, and $$B = \begin{bmatrix} 2 & 0 \\ 1 & 7 \\ \end{bmatrix}$$?

Solution: Here both the matrices A and B are of $$2 \times 2$$ order hence the addition of these matrices is possible. $$A + B = \begin{bmatrix} 3 + 2 & 4 + 0 \\ 5 + 1 & 6 + 7 \\ \end{bmatrix}$$ $$A + B = \begin{bmatrix} 5 & 4 \\ 6 & 13 \\ \end{bmatrix}$$ The order of final added matrix (A + B) is also same $$(2 \times 2)$$.

Example(2): The number of boys and girls students studying three subjects Mathematics, English, and Hindi, in two different schools A and B as given below. How many boys and girls studying Mathematics, English, and Hindi in both schools together? $$A = \begin{matrix} & Boys & Girls \\ Math & 75 & 60 \\ English & 50 & 70 \\ Hindi & 60 & 80 \\ \end{matrix}$$ $$B = \begin{matrix} & Boys & Girls \\ Math & 40 & 45 \\ English & 50 & 40 \\ Hindi & 60 & 70 \\ \end{matrix}$$

Solution: Let's write down the number of boys and girls studying Mathematics, English, and Hindi in matrix form. $$A = \begin{bmatrix} 75 & 60 \\ 50 & 70 \\ 60 & 80 \\ \end{bmatrix}$$ $$B = \begin{bmatrix} 40 & 45 \\ 50 & 40 \\ 60 & 70 \\ \end{bmatrix}$$ Here the order of both the matrices is the same $$3 \times 2$$ hence the addition of these matrices is possible. $$A + B = \begin{bmatrix} 75 + 40 & 60 + 45 \\ 50 + 50 & 70 + 40 \\ 60 + 60 & 80 + 70 \\ \end{bmatrix}$$ $$A + B = \begin{bmatrix} 115 & 105 \\ 100 & 110 \\ 120 & 150 \\ \end{bmatrix}$$ The number of boys studying Mathematics, English, and Hindi in both the schools' A and B together, are 115, 100, and 120 respectively.

The number of girls studying Mathematics, English, and Hindi in both the schools' A and B together, are 105, 110, and 150 respectively.

Example(3): If $$A = \begin{bmatrix} 4 & 3 & 2 \\ 1 & 5 & 6 \\ \end{bmatrix}$$, and $$B = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \\ \end{bmatrix}$$ then find the (A + B)?

Solution: Here the order of matrix A is $$2 \times 3$$, and the order of matrix B is $$3 \times 2$$. Hence these matrices cannot be added. The order of both the matrices must be the same.

Example(4): If $$A = \begin{bmatrix} 5 & 6 & 7 \\ 2 & 0 & 1 \\ 0 & 1 & 3 \\ \end{bmatrix}$$, and $$B = \begin{bmatrix} 1 & 4 & 6 \\ 2 & 4 & 0 \\ 1 & 2 & 5 \\ \end{bmatrix}$$ then find the (A + B)?

Solution: Here the order of both the matrices is the same $$3 \times 3$$ hence the addition of these matrices is possible. $$A + B = \begin{bmatrix} 5 + 1 & 6 + 4 & 7 + 6 \\ 2 + 2 & 0 + 4 & 1 + 0 \\ 0 + 1 & 1 + 2 & 3 + 5 \\ \end{bmatrix}$$ $$A + B = \begin{bmatrix} 6 & 10 & 13 \\ 4 & 4 & 1 \\ 1 & 3 & 8 \\ \end{bmatrix}$$