How is each entry in a matrix product calculated?

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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.

To calculate the product of two matrices, $A$ and $B$ , the number of columns in $A$ must equal the number of rows in $B$ . The resulting matrix $C$ is formed by calculating each entry $c_{i j}$ , which represents the element in the $i$ th row and $j$ th column of the product.

To find $c_{i j}$ , you perform a dot product operation: multiply the first element of the $i$ th row of $A$ by the first element of the $j$ th column of $B$ , then multiply the second element of the row by the second element of the column, and continue this process for all corresponding elements. The sum of all these individual products equals the value of $c_{i j}$ .1

In mathematical notation, if $A$ has $m$ columns and $B$ has $m$ rows, the entry is calculated as:
$c_{i j} = \sum_{k = 1}^{m} a_{i k} b_{k j}$

The resulting matrix $C$ will have the same number of rows as $A$ and the same number of columns as $B$ .1