Check that the number of columns in the first matrix equals the number of rows in the second matrix
Let the first matrix be A and the second matrix be B
For each entry in the product matrix, take the dot product of one row from A and one column from B
Multiply corresponding elements in the chosen row and column
Add the products together
Place the result in the matching position of the new matrix
Repeat for every row of A and every column of B
Write the final matrix with all computed entries