Definition of a Matrix.
The aim of this supplementary lecture is to help you understand
the “practical meaning” of the matrix product. We discuss the
related problems in homework B1. We as sume that you are
familiar with the usual matrix ope rations already . Compare this
lecture with Lecture 3, where we did similar work, without
bringing in the formalism of matrices. The fol common -m.html">lowing is based on
WHS problem 12 on B1 (common). Suppose that Bill holds
12, 12, 15, 16 stocks respectively of IBM, Google, Toyota and
Texaco. Suppose that Jim holds 15, 11, 16, 17 stocks of same
companies respecyively. If the prices of these stocks are 5, 1, 2, 3
respectively, then we use matrices to organize the information and
calculate.
Organization of the data
•We begin by building a matrix to record the above data as
follows. We have added in row and column titles for
understanding, bt they do not take part in matrix operations .
•The holding Matrix.
•The price matrix.
•We respectively write A = A2×4 and B = B4×1 for the two
matrices .
•Note that we have
•And we have:
The calculation clearly shows that it is giving us the values of
the holding of Bill and Jim respectively. This answers the
questions.
Another Example.
•The following is based on WHS problem 8 on B1 (common).
•we are given matrices (with informative headers):
and
Meaning of the product.
•As before, if we name the matrices A = A4×4 and B = B4×1,
then AB has type 4 × 1. Its four rows correspond to the four
rows of A, thus belong to the indicated states.
•The resulting column gives the profit totals.
•Thus the (2, 1) entry of AB gives the profit made in OH(IO)
and is equal to: 17 · 19 + 25 · 11 + 24 · 15 + 23 · 9 = 1165.
•Similar inter pretations can be made by using the meanings of
the rows and columns of the matrix.
•Various examples in Chapter 2.4 should be reviewed to
understand this concept.
