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How are matrices added together?

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vector parallelogram for addition and subtraction — linear algebra One method of adding and subtracting vectors is to place their tails together and then supply two more sides to form a parallelogram. The vector from their tails to the opposite corner of the parallelogram is equal to the sum of the original vectors. The vector between their heads (starting from the vector being subtracted) is equal to their difference.

In linear algebra, two matrices can be added together only if they have the same number of rows and the same number of columns.1

If you have two $m \times n$ matrices, $A$ and $B$ , their sum $S = A + B$ is also an $m \times n$ matrix. Each element $s_{i j}$ of the resulting matrix $S$ is calculated by adding the elements in the corresponding positions of $A$ and $B$ :

$s_{i j} = a_{i j} + b_{i j}$

In other words, you simply add the value located in the $i$ th row and $j$ th column of matrix $A$ to the value located in the same $i$ th row and $j$ th column of matrix $B$ to find the corresponding element in the new matrix.1