ITI 1121 Lab 8 Introduction to matrices solution


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• A matrix is a two dimensional rectangular grid of numbers:
• The dimensions of the matrix are the numbers of rows and columns (in
the above case: row dimension 3, column dimension 3).

• A value within a matrix is referred to by its row and column indices, in
that order.

– Math: number rows and columns from 1, from upper left corner
• In math notation, M1,2 = 2

– In Java, matrices are implemented via 2D Arrays and indices start
from 0, as they do in (1D) arrays.
• Thus, M[0][1] = 2

7 8 9
4 5 6
1 2 3
Matrix element processing

• To visit every element of an array, we had to
use a loop.

• To visit every element of a matrix, we need to
use a loop inside a loop:
– Typically the outer loop goes through each row of
a matrix
– And the inner loop goes through each column
within one row of a matrix.

Matrix example
Write a Java program that finds the sum of the
upper triangle of a square matrix (i.e. the
diagonal and up).
1 4 5 3 2
6 3 6 4 6
M = 4 3 6 7 2
3 4 2 2 4
2 3 8 3 5
How do we know if an
element of a square
matrix is on or above
the main diagonal?
row_index <=
0 1 2 3 4

Matrix element processing
Spend some time studying more examples of
processing 2D-array from your textbook
(included in this lab processing2DArrays.pdf

Exercise 1: Algebra – Matrix Transpose

• Write an Java method, called transpose, that takes a matrix of
integers A as input and transposes the matrix to produce a new
matrix AT. Write your solution inside clearly indicated space in transpose of the matrix is such
that element arc in the original matrix will be element aT
cr in the
transposed matrix.

The number of rows in A becomes the
number of columns in AT, and the number of columns in A
becomes the number of rows in AT.

• For example:

⎤ ⎢

⎡ = 6


Adjacency Matrix
• Escape Airlines has flights between certain cities.
The flights and their costs can be represented as a
graph in which an edge between city x and city y with
a weight (label) of w means Escape Airlines has a
flight between x and y costing w dollars.

Ottawa (0)
Madrid (4)
Toronto (3)
Paris (2)

Matrix Representation
• This graph can be represented with an adjacency matrix. There
is a row and a column for each city, and cost[x][y] is the cost of
a flight from x to y if one exists and is infinity (∞) if there is no
such flight.

• Here, “infinity” is actually a very large number, greater than any
– In Java: a predefined constant is available for the largest
possible integer: Integer.MAX_VALUE
0 500 450 300 700
500 0 900
cost 450 0 400
300 900 400 0 300
700 300 0
⎡ ⎤
⎢ ⎥ ∞ ∞
= ∞ ∞
⎣ ⎦ ∞ ∞

Exercise 2: Finding the
Affordable Direct Flight

• Suppose you live in one of Escape’s cities and have $d to spend.
Write a Java method, called cheapDirectFlights that returns an
array of the cities you can afford to fly to directly.

• What do you know (Givens: input your method)
1. The city where you live.
2. The cost of flight between two cities.
3. The amount you can spend.
• What you want (Result: output of your method)
– An array of cities that can be visited.

• Idea:
– First, find the cities that can be visited.
– Then, create an array of the right size.
– Finally, place cities that can be visited in the array. Your
method should return that array, unless no city can be visited
in which case null should be returned

Use the provided Java
• To solve the exercise, you are provided with a file
called You
should open this file work with it i.e. all your code for
this exercise should go inside this file in the clearly
indicated spaces.

Programs with more than one
• A program may have more than one class. If you save
all classes in a program in one directory/folder, any
class may call a public method in any other class in
the same directory.
• When a (static) method is called from another
class, use the name of the class with the dot

– For example, if we include class Library that has
a method called aMethod, then another class that
is in the same directory, for example one a class
that contains your assignment, may call that
method with call Library.aMethod( );

Library Classes
• Instead of putting all our methods in the same class
as main (the class that contains our program) it is
often better to separate them into coherent groups
and put each group in a class of its own.

• These classes will not be programs – they have no
main method. Each will be a small library of methods
that can be used by other methods.
• Such classes can be compiled on their own but cannot
be run as standalone programs.

1. Open and study to two provided libraries: and mySearchSortLibrary

2. Write a short Java program that creates and array
{5.0, 4.4, 1.9, 2.9, 3.4, 3.5} and then uses the
methods in the two provided libraries to
– print the array,
– sort the array,
– print it again
– and finally look for an element in that array both with
linear search and binary search.

Exercise 3: Using your own libraries