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ICS4U - Lab 1
Objective
This lab is designed to give you experience in creating and using objects.

Part 0 – TwoByTwoMatrix Class – TwoByTwoMatrix.java

You are to create a class called TwoByTwoMatrix containing fields and methods that could be used by a main method in another class to manipulate 2 by 2 matrices in various ways. A TwoByTwoMatrix is defined by the following fields:

class TwoByTwoMatrix
{
double a; // Top left element
double b; // Top right element
double c; // Bottom left element
double d; // Bottom right element

This corresponds to a matrix that looks like this: --- where a, b, c and d are real numbers.

You are to write the following constructors:

●// Creates a TwoByTwoMatrix assigning 0 to each element
public TwoByTwoMatrix ()

●// Creates a TwoByTwoMatrix with parameters giving initial
// values of the elements
public TwoByTwoMatrix (double a, double b, double c, double d)

You are to give the class functionality by writing the following instance methods:
(a)A toString method that will return a String of the TwoByTwoMatrix . For example, if the matrix is , the toString() method should return . Round to a maximum of 3 decimal places but ONLY for display purposes - meaning, do NOT change the values of the implicit matrix.

(b)A method called determinant that returns the determinant of this TwoByTwoMatrix as a double. See this video - https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:matrices/x9e81a4f98389efdf:determinant-of-2x2-matrix/v/finding-the-determinant-of-a-2x2-matrix

(c)A method called plus that returns the sum of this TwoByTwoMatrix and another as a TwoByTwoMatrix.

(d)A method called minus that returns the difference of this TwoByTwoMatrix and another as a TwoByTwoMatrix. (implicit minus explicit)

(e)A method called multiply that returns the product of this TwoByTwoMatrix and another as a TwoByTwoMatrix. https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:matrices/x9e81a4f98389efdf:multiplying-matrices-by-matrices/v/multiplying-a-matrix-by-a-matrix

(f)A method called scalarMultiply that takes a double parameter and scalar multiplies it with this TwoByTwoMatrix. This method should not return anything. See video: https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:matrices/x9e81a4f98389efdf:multiplying-matrices-by-scalars/v/scalar-multiplication

(g)A boolean method called isIdentityMatrix that returns true if this matrix is an identity matrix and false otherwise. If you get the identity matrix to 3 decimal places, then you may assume that it is the identity matrix. Use the roundThreeDecimals method to round each value to 3 decimal places (but only for checking if it is the identity matrix. ie. do not change the values in the Matrix).

(h)A boolean method called isInvertible that returns true if this matrix has an inverse and false otherwise. The inverse of a 2 by 2 matrix does not exist if and only if its determinant is 0. If the determinant is 0 after rounding to 3 decimal places, you may consider that to be 0.

(i)A boolean method called isInverse that takes another TwoByTwoMatrix as a parameter and returns true if this matrix and the other matrix are inverses of one another and false otherwise. Matrices A and B are inverses of one another if AB = BA = I where I is the 2 by 2 identity matrix. If you get the identity matrix to 3 decimal places, then you may assume that it is the identity matrix.

(j)A boolean method called equals that takes another TwoByTwoMatrix as a parameter and returns true if the matrices have the same elements in the same positions and false otherwise. Since the values are doubles and could be long decimals, this method will return true if all the elements are the same to 3 decimal places. You will write a class helper method to this effect (see method (m) below)

(k)A method called inverse that returns as a new TwoByTwoMatrix the inverse of the implicit TwoByTwoMatrix if it exists and throws an exception if it does not. The inverse of a 2 by 2 matrix does not exist if and only if its determinant is 0. See FAQ note.

(l)A method called transpose that returns as a new TwoByTwoMatrix the transpose of the implicit TwoByTwoMatrix.

You must write the following class method:

(m)A method called roundThreeDecimals that takes a double parameter and returns the input rounded to three decimal places also as a double.

You should test your program thoroughly and systematically using a main method in another class. Your main method will not be marked. Be sure that your methods are titled exactly as written in this document as your methods will be tested with my own main method. Marks will be deducted if I need to change your program in order to make it work with mine.

Question:
G) For the isIdentityMatrix question I don't understand the decimal part "If you get the identity matrix to 3 decimal places, then you may assume that it is the identity matrix. Use the roundThreeDecimals method to round each value to 3 decimal places (but only for checking if it is the identity matrix. ie. do not change the values in the Matrix)."

For example, let's say I send your method the matrix
[1 0]
[0 1]
Your method should return true since this is the identity matrix.
If I send your method the matrix
[1 2]
[3 4]
Your method should return false as this is not the identity matrix.
Now, what if I send your method the matrix
[1.0003 0.0001]
[0.0001 1.00002]
If you don't round these values, your method will return false because you are comparing with exactly 1, 0, 0 1. But in this case, this is "close enough" to be the identity matrix. So in this case, the rule is to round each value to 3 decimal places. Rounded to 3 decimal places the first value 1.0003 will round to 1.000 and then when you compare to 1, you would get a true for that component. Same with the other 3 components once rounded.
In other words, this last matrix should cause your method to return true. This same idea of “rounding before checking” should be used in the equals() method as well - part j.

Question:
How do I throw an exception if the inverse Matrix doesn’t exist? This is for part k)
If the inverse of the given matrix does not exist, i.e. the determinant is 0, then you can throw an exception like this:
throw new RuntimeException(“Inverse Matrix does not exist”);
This will allow your method to compile because if you throw an Exception, you don’t have to return a TwoByTwoMatrix in this situation.
Also if this line executes, your program will immediately stop running and display this error (much like an ArrayIndexOutOfBoundsException)
Question:
For the inverse method, part k, would a determinant of 0.00001 be considered to be 0 for the purposes of this method?