3. Multiplication of Matrices

We have three recipes for breakfast foods. Each recipe feeds three people. The ingredients are as follows:

Pancakes: 2 cups baking mix, 2 eggs, and 1 cup milk.
Biscuits: cups baking mix and cups milk.
Waffles: 2 cups baking mix, 1 egg, cups milk, and 2 tablespoons vegetable oil.
Let's write this in the form of a labeled matrix so that it is easier to read.

If we want to feed 6 people instead of 3, what do we need to do? We double each recipe. That means we need twice as much of each ingredient, so we multiply every element of the matrix by the number 2.

When we multiply a matrix by a real number, we call the real number a Scalar and call the operation scalar multiplication. Scalar multiplication consists of multiplying each element of a matrix by a given scalar. We use the terms scalar and scalar multiplication because, in abstract algebra, we often have the need to consider more general scalars than real numbers. However, in this book, we restrict our attention to scalars that are real numbers.

Definition 3.1 If c is a real number and A is a matrix whose (i,j)^th element is a_ij, then the scalar product cA is the matrix whose (i,j)^th element is ca_ij.

If we want to feed 3 people pancakes, 12 people biscuits, and 9 people waffles, how much baking mix will we need? We need to make one batch of pancakes, 4 batches of biscuits, and 3 batches of waffles. Let's represent this with the row vector

We could have written this as a column vector instead of a row vector, but a row vector will be useful in later problems. The vector describing how much baking mix we need is the first column of R. We will call it a. Therefore,

We need cups of baking mix. The process by which we found the number of cups of baking mix needed is called finding the inner product of two vectors.

Definition 3.2 The inner product of n-vectors x and y, denoted by where n is the dimension of the vectors.

Remark 7 Some people refer to the inner product as the dot product and denote it x·y.

Now let's multiply our row vector, s, by our recipe matrix, R.

We need 17 cups of baking mix, 5 eggs, 8 cups of milk, and 6 tablespoons of oil.

There are several interesting things to notice about matrix multiplication. We multiplied a 1 by 3 matrix by a 3 by 4 matrix and got a 1 by 4 matrix. This pattern will always hold when we multiply. The middle numbers must be the same (like the threes were in this case), when we multiply matrices. The resulting matrix will always have the dimensions of the outside numbers (1 by 4 in this case) when multiplication is defined. The following picture expresses the requirements on the dimensions:

The label, total by ingredients, is meaningful because foods was the label for the inside dimensions of both matrices that we multiplied.

Let's also look closely at how we multiply the matrices because we will multiply matrices with larger dimensions later. This is a hands on activity. Take your left pointer finger and place it at the beginning of the first row of the first matrix (the only row we have in this case). Take your right pointer finger and place it on the first number of the first column of the second matrix. Multiply the two numbers to which you are pointing. Each time you move, your left hand will go across the row, and your right hand will go down the column. When you reach the end of the row and column, add the numbers you have obtained from the multiplications. This number goes in the first row and first column of your product matrix. This is the same as taking the inner product of the first row of the first matrix and the first column of the second matrix. Now you can move to the first row, second column doing the same thing. This number will go in the first row, second column of your product matrix. In short, position ij of your product matrix consists of the inner product of the i^th row of your first matrix and the j^th column of the second matrix. This is a lot easier to do than it is to describe! Your left hand will move across and your right hand will move down. Do this for every row and column combination to get your product matrix. No, you are not too old to do this. A lot of college students multiply matrices this way. After you do this enough times, your hands will not let you do it incorrectly ever again. This picture depicts the motions necessary to find a product:

Definition 3.3 Consider the m by p matrix A and the p by n matrix B. By the matrix product A times B, we mean the m by n matrix whose (i,j)^th element is the inner product of the i^th row of A with the j^th column of B.

Look at the dimensions of our matrices and the labels we have put on them. THINK about what dimensions and labels should be on our product matrix. This will tell you how we should organize the data about the cost of each food. Write down your product matrix before you read further. Also write down each step of the multiplication and addition that you do to find the product matrix. Compare that to the matrix products that follow. Look carefully at where each number that you used appears in the matrix.

Let's call our cost matrix C. Remember that since the dimensions of C are 4 by 1, C could also be considered to be a column vector of dimension 4.

Now let's multiply to find out how much the ingredients of each recipe cost.

This gives us the cost of the ingredients needed for each food to feed three people. Biscuits are cheaper than pancakes which are slightly cheaper than waffles.

Remember to use your fingers the way we discussed earlier to remember which numbers to multiply. Let's look at the dimensions and labels for this example. The dimensions of matrix R are 3 by 4 and the dimensions of C are 4 by 1, so R*C is 3 by 1. The labels also tell us that we set up the product correctly. We have food by ingredients multiplied by ingredients by cents to get food by cents. This is what we want!

What if we also want to know the calorie content of each recipe? If we know the calorie content of each ingredient, we can find the number of calories in each recipe. There are 510 calories per cup of baking mix, 70 calories in each egg, 90 calories in a cup of milk, and 120 calories in a tablespoon of oil. Look at the labels for matrix R and for the product matrix to decide how to organize this information. We will put this in a 4 by 1 matrix (also called a column vector of dimension 4) and name it K. To find out how many calories are in each recipe, we multiply R*K.

Now we know that there are 1250 calories in a recipe of pancakes, 1215 calories in a recipe of biscuits, and 1450 calories in a recipe of waffles for three people. Therefore, biscuits have the fewest calories, pancakes are in the middle, and waffles have the most.

What would we do if we want to know both the cost of each food and the number of calories? We could use matrix multiplication twice like we did above, but we also have the ability to set up only one matrix multiplication to find both pieces of information. We can append K to C to form a single matrix. Let's call it F.

Now let's multiply R*F to find the information we want.

Would we have been able to multiply F*R? No, the dimensions are wrong because matrix multiplication is defined only if the "inside dimensions" agree. What happens when you try to multiply these matrices? You run out of numbers in the row and column at different times. This should alert you to the fact that something is wrong.

Remark 8 A shortened way of writing R*F is RF. When there is no sign between two matrices or two sets of parentheses, it is implied that you should multiply.

Definition 3.4 An identity matrix is a square matrix with ones along the main diagonal and zeros elsewhere.

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