Brooklyn College MDM4U Mathematics of Data Management

1.7 Problem Solving With Matrices

Investigate & Inquire: Matrix Multiplication

  1. Calculate the number of points for each team in the Northeast Division using the above tables. Explain your method.
  1. a) Represent the team standings as a 5 x 4 matrix, A.

b) Represent the points system as a column matrix, B.

  1. Describe a procedure for determining the total points for Ottawa using the entries in row1 of matrix A and column 1 of matrix B.
  1. How could you apply this procedure to find the points totals for the other four teams?
  1. Represent the total points for each team as a column matrix, C. How are the dimensions of C related to those of A and B?
  1. Would it make sense to define matrix multiplication using a procedure such that A x B = C? Explain your reasoning.

Multiplication of Matrices

General: For example:

Identity Matrices

Form:

with entries of 1 along the main diagonal and zeros for all other entries.

With dimension n x n.(square matrix)

Is represented by .

for any matrix A.

Inverse Matrices A-1

For most square matrices, there exists an inverse matrix A-1with theproperty that

. Note that

Since

However, if ad=bc, then A-1does not exist since it would require dividing by zero!!

b)

Multiplying the coded message by this key gives

Network Matrices

Transportation and communication networks can be represented using matrices, called network matrices. Such matrices provide information on the number of direct links between two vertices or points (such as people or places). The advantage of depicting networks using matrices is that information on indirect routes can be found by performing calculations with the network matrix.

To construct a network matrix, let each vertex (point) be represented as a row and as a column in the matrix. Use 1 to represent a direct link and 0 to represent no direct link. A vertex may be linked to another vertex in one direction or in both directions. Assume that a vertex does not link with itself, so each entry in the main diagonal is 0. Note that the network matrix provides information only on direct links.

Homework Practice: Pg74 #1 2 4 5 7 12 13 15 16

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