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Chapter 2
The Coefficient Matrix

From the fact that zero times any number is zero presents a situation where a system of linear equations cannot be solved. Two simple examples show this:
[1]
0x = 0
and
[2]
0x = 5
Every number is a solution to equation [1]. No number is a solution to equation [2]. Equation [2] states that 0 = 5.

Similarly, the simultaneous equations
[3]

2x + 3y = 11
10x + 15y = 55
cannot be solved. The second equation does not add any more definitive information that the first equation. Any pair of numbers for x,y will satisfu both equations. The equations [4]
2x + 3y = 11
10x + 15y = 60
cannot be solved for a different reason. The second equation contradicts the first. Dividing the second equation by 5 produces the impossibility that the expression 2x + 3y is equal to two different numbers, namely 11 and 12. These two situations , all numbers are solutions, and no numbers are solutions, present the two extremes that may happen when trying to solve simultanous equations. Behind the scenes, zero plays a definitive role.

To handle the three situations, no solutions, one unique solution, infinitely many solutions, it is convenient to "split" the matrix of the equations into two associated submatrices.
The splitting of the matrix in Fig 1a produces two matrices in Fig 1b and Fig 1c. The matrix in Fig 1b is a 2x2 matrix. It is a square matrix. It is called the coefficient matrix of the equations:
[5]

3x + 4y = 10
2x + 5y = 9
The matrix in Fig 1c is a 2x1 matrix and is a column matrix. For lack of a standard name, it is called here the right column matrix of values of the equations [5] or simply right column matrix.

Zeros play an important role in matrices. Consider the coefficient matrix and the right hand column matrix

	6   0   0   0   0	42
	0   1   0   0   0	 4
 	0   0   7   0   0	 7
	0   0   0   4   0	12
	0   0   0   0   2	 4
shows a coefficient matrix separated from the right column matrix. It is very easy to find the solutions 7, 4, 1, 3, 2.. Divide the first row (in each matrix) by 6, the third by 7, etc. The result is
[6] 
	1   0   0   0   0	7
	0   1   0   0   0    	4
	0   0   1   0   0	1
   	0   0   0   1   0	3
  	0   0   0   0   1	2
Both of these matrices are diagonal matrices. They have integers = 0 everywhere except on the diagonal. ( There may be zeros there also.) This is the main diagonal, from upper left to the lower right. There is another diagonal, from lower left to upper right.

The diagonal matrix [6] with all 1's on its main diagonal is called an identity matrix. Motivation for the name "identity" will be given later. The symbol I5 can be used to designate the 5x5 identity matrix. The symbol In can be used to indicate the nxn identity matrix. However, there seldom is any confusion when the subscript n is omitted.

Included among the computer programs is a program (makeequ = make equations). It askes for the number of unknowns and the solution to them. Then all coefficients are requested, line by line. The program prints out the coefficient matrix and the right hand column of values. For example, if the number of unknowns is 2, and the solution is 2 and 1, and the coefficients are 2 3 for the first row and 4 5 for the second row, the computer outputs (only the matrix of the equations)

                                    2   3     7		(2x + 3y = 7)
                                    4   5    13		(4x + 5y = 13)
It has calculated the column vector
[7] 
			7
			13
Suppose now [7] is to be a solution to 
		6w + 7t  = ?
		8w + 9t  = ??
The computer will output: 
		6   7    133		(6w + 7t = 133)
		8   9    173		(8w + 9t = 173)
Then what has happened to the column vectors is: 
		
		2  =>	7   =>	133
		1   	13	173





The coefficient matrix can be used to determine if a system of linear equations can be solved.

This is done by evaluating a square matrix to get a specific number (in these discussions, an integer) that will determine the solvability. If the number is zero, then no unique solution exists, if not zero then there is a unique solution (the "normal" situation). One way to find this number is to try to solve the equations by converting the matrix of coefficients into the matrix of the form