We now explore some other properties of matrices as mathematical objects. Suppose that we seek the nutrition information for banquet of 40 breakfasts and 10 lunches. We can do compute this information in one of two ways. One way is to first compute the total number of eggs, potatoes, pork and pancakes that we need first. So, we multiply 40 breakfasts, and each breakfast has two legs and one potato giving a total of 80 eggs and 50 potatoes. The 50 potatoes includes also the 10 lunches. Once we have the total number of portions we can multiply by the nutrition information. So, for instance, the number of calories in this combination would be 100 calories for each egg times 80 eggs plus 340 calories for each potato times the number of potatoes etc. Giving a total number of calories of 26,700 and similarly for the all these other entries or we can first compute the nutrition information for a single meal. So, we previously computed that at breakfast consisting of two eggs and one potato has 540 calories and one lunch consisting of one potato, one bacon and one pancake consists of 510 calories. We then multiply by the total number of breakfasts and lunches. In either case, we get the same answer. We can either multiply the total number of meals times the portions in each meal to get the total number of portions we will have to order then we multiply by the nutrition information for each ingredient. Or we compute the nutrition information for each meal in total that is the nutrition for the entire breakfast than in the nutrition for the entire lunch and then we multiply by the number of breakfasts and number of lunches. The fact that you get the same answer either way is called the associative rule. This works because we've arranged it that all the dimensions are compatible. So the total number of meals is a one by two matrix that is the 40 breakfasts and 10 lunches. The nutrition for the portions for each meal is two by four. We have two meals and four ingredients. The nutrition information is four by seven. We have four ingredients and seven pieces of nutrition information. We can see that the inner dimensions, the two and the two get summed out and the four and the four get summed out. The result is a one by seven matrix. That is at the end the outer dimensions are the ones that survived at the end. Another rule is the distributive rule. Which can be written this way. You can think of this as follows. In words, we can compute the nutrition information for the two banquets in two different ways. First we can compute the nutrition information for each banquet and then add them together. So we compute the nutrition information for the breakfast and for the lunch, for two banquets because we have to banquets. For banquet number one we have some number of meals times the nutrition information for each meal, and for banquet number two we have the number of meals times the same nutrition information because we're using the same ingredients. We could come first the total number of portions for the two banquets combined and then apply the nutrition information. The result is the same either way. This is called a distributive rule. That first we can multiply this and then add the results or we can multiply the left hand side and then add the left hand side together and then multiply by this final C.
