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 Matrices of the same dimensions are added by adding corresponding elements.
For instance, aij corresponds to bij because they both lie in the ith row and jth column of their respective matrices. Therefore, we would add, aij + bij 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 Matrices of the same dimensions are subtracted by subtracting corresponding elements.Suppose Y represents the wins, losses, and ties for these teams for the entire season (regular season and the playoffs together). Consider the following data
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 Remember, to add (or subtract) matrices, add (or subtract) corresponding elements.The addition property of zero for real numbers tells us that r + 0 = 0 + r = r. There is also an addition property of zero for matrices which states that A + 0 = 0 + A = A where 0 represents the zero matrix of the same dimensions as A.
Definition A zero matrix is a matrix which has the number 0 for each of its elements.Remark 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
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