2. Addition of Matrices

If the Cardinals won 7 games in the first half of the regular season and won 8 in the second half, how many games did they win during the regular season? You know that the answer is 15 because 7 + 8 = 15. The Eagles lost 8 games in the first half and lost 6 in the second half of the season. How many games did the Eagles lose all season? They lost 14 games. We know how to answer these questions using real numbers because we have represented our data by real numbers, and addition, subtraction, and multiplication are all defined and well-known operations for real numbers. However, how would we add when our information is represented by matrices? Let the matrix A represent the statistics from the first half of the season, and let the matrix B represent the statistics from the second half of the season.

Look carefully at how you answered the questions above. Then look at where those numbers appear in the matrices. How would you add A + B?

Take time to think before reading further!

Definition 2.1 Matrices of the same dimensions are added by adding corresponding elements.

For instance, a_ij corresponds to b_ij because they both lie in the i^th row and j^th column of their respective matrices. Therefore, we would add, a_ij + b_ij to obtain the (i,j)^th element of A + B.

Think about the similarities between addition and subtraction. How do you think matrices are subtracted?

Definition 2.2 Matrices of the same dimensions are subtracted by subtracting corresponding elements.

How would you find the number of wins, losses, and ties for the playoffs? We would subtract the number of wins, losses, and ties for the regular season from the number of wins, losses, and ties for the entire season.

Remark 5 Remember, to add (or subtract) matrices, add (or subtract) corresponding elements.

Definition 2.3 A zero matrix is a matrix which has the number 0 for each of its elements.

Remark 6 We say "a" zero matrix instead of "the" zero matrix because for different pairs of dimensions, we have different zero matrices. However for a given pair of dimensions, the zero matrix is unique because zero is unique in the real number system. It is usual to merely say the zero matrix and not refer to its dimensions when no confusion can arise.