Preface
Chapter 2
Chapter 3

Additional Material for this Chapter
Exercises for this Chapter
Answers to Exercises

Chapter 1
Matrices of Linear Integer Equations

In linear equations the unknowns are not raised to any powers, nor do they involve roots, trig functions, logs, etc. The simplest linear equation is in one unknown, for example:
                                                 3x = 6
In this situation x denotes the unknown, 3 is the coefficient of that unknown, 3x is a linear expression containing the unknown, and 6 is the (right) value of that expression.. The equation is easily solved to give x=2. Actually the letter x plays a minor role. The equation   3u=6    represents the same equation, only using a different letter for the unknown. The solution is u=2. In one case one linear equation is in x, while the other is in u. The impmortant parts of the equation are the coefficient and the right value::

MAT    3    6
Probably the symbol EQU should be used to indicate that the line is a shorthand for equation, without an unknown and equality sign. But MAT will be used to mean equation. MAT has the meaning of matrix.

In these chapters all numbers as coefficients will be integers. However, solutions may be quotients of integers (rational numbers). The solution to 3x=7, written as

MAT     3    7
is the rational number 7/3
MAT     1   7/3.
No real numbers will be used in these discussions. Instead of writing 2.3333... as a solution to the equation, the fraction 7/3 will be used. Using whole integers and quotients of integers will avoid some difficulties using decimal real numbers, difficulties beyond the scope of this intuitive work.

It is more interesting to consider more than one linear equation. The word "simultaneous" implies two or more components of an event happening together, and all should be considered together. In simultaneous equations the same unknowns (letters) are used. The two simultaneous linear equations in x and y
[1]

3x + 4y = 10
2x + 5y = 9
can easily be solved: x = 2, y = 1. (For a process of solving these equations click here.) The equations
3u + 4v = 10,
2u + 5v = 9
have the solution u = 2, v = 1. Other letters can be used for the unknowns.

The essential part of of these equations are the numbers 10 and 9 and the coefficients of the unknowns. The notation
[2]


contains all essential information without any loss. This table of six numbers is called the matrix of the equations. The size is 2x3 (2 rows , 3 columns) The top row = row 1 is (3 4 10) and the middle column = column 2 contains the numbers 4 5.

The matrix of equations is a short-hand for writing down those equations. Any legal operation on any equation to produce a new equation does not change the solution. For example, for equations [1] above if the top equation is multiplied by 5 and the bottom equation is muyltiplied by 4, a new set of equations is obtained:

15x + 20y = 50
8x + 20y = 36
[2] above if the top row is multiplied by 5 and the bottom row is multiplied by 4, a new new matrix is obtained:
                                                
Subtraction of the equations and rows in the matrix leads to the solution of the equations and the corresponding matrix:
                                                 x=2
                                                 y=1

The solution x=2, y=1 can be written in the form of simultaneous equations in x,y:
                                                 1x + 0y = 2
                                                 0x + 1y = 1
This equivalent form is better for comparison with the matrix of the equations.

For the details of this process of solving [1] for x,y both by ordinary algebra with equations and corresponding row methods of the matrices of equations click here.

Three rules of Algebra for handling simultaneous liner equations are useful here:
(A1) In any equation, every number may be multiplied by the same non-zero number.
(A2) Any two equations may be added together and this sum may replace either equation (but not both).
(A3) Any two equations may be interchanged.

The corresponding rules for handling Matrices of equations are:
(M1) In any row, every number may be multiplied by the same non-zero number.
(M2) Any two rows may be added together and this sum may repalce either row (but not both).
(M3) Any two rows may be interchanged.

Either set of the above rules can be used to solve the three simultaneous equations directly by algebra or by operations on the rows of their matrix:
                                                 2x + 3y - 5z = 7
                                                 3x + y = 15
                                                 -y + z = -1
can be solved to find x=4, y=3, z=2. This solution can be represented as:
                                                 1x + 0y + 0z = 4
                                                 0x + 1y + 0z = 3
                                                 0x + 0y + 1z = 2

For persons using pencil and paper the solution (if any exist) of simultaneous linear equations is found by using elementary algebra. But for persons using a computer, a program can be created to find the solution (if any exist) using matrices of equations. For such a program click here (If somehow the equations are "abnnormal" and cannot be produce ordinary solutions the program can output an error message explaining the reason. It is also possible to think up a collection of equations that may produce a solution that does not satisfy the equations! A discussion of such "abnormal" situations are discussed in the next chapter (2).