28 CHAPTER 1. PRELIMINARIES
1. Switch two rows.
2. Multiply a row by a nonzero number.
3. Replace a row by a multiple of another row added to it.
It is important to observe that any row operation can be “undone” by another inverserow operation. For example, if r1, r2 are two rows, and r2 is replaced with r′2 = αr1 + r2using row operation 3, then you could get back to where you started by replacing the row r′2with −α times r1 and adding to r′2. In the case of operation 2, you would simply multiplythe row that was changed by the inverse of the scalar which multiplied it in the first place,and in the case of row operation 1, you would just make the same switch again and youwould be back to where you started. In each case, the row operation which undoes whatwas done is called the inverse row operation.
Example 1.11.3 Give the complete solution to the system of equations, 5x+10y−7z = −2,2x+ 4y − 3z = −1, and 3x+ 6y + 5z = 9.
The augmented matrix for this system is 2 4 −3 −1
5 10 −7 −2
3 6 5 9
Multiply the second row by 2, the first row by 5, and then take (−1) times the first row andadd to the second. Then multiply the first row by 1/5. This yields 2 4 −3 −1
0 0 1 1
3 6 5 9
Now, combining some row operations, take (−3) times the first row and add this to 2 timesthe last row and replace the last row with this. This yields. 2 4 −3 −1
0 0 1 1
0 0 1 21
.
Putting in the variables, the last two rows say z = 1 and z = 21. This is impossible sothe last system of equations determined by the above augmented matrix has no solution.However, it has the same solution set as the first system of equations. This shows there is nosolution to the three given equations. When this happens, the system is called inconsistent.
This should not be surprising that something like this can take place. It can even happenfor one equation in one variable. Consider for example, x = x+1. There is clearly no solutionto this.
Example 1.11.4 Give the complete solution to the system of equations, 3x − y − 5z = 9,y − 10z = 0, and −2x+ y = −6.
The augmented matrix of this system is 3 −1 −5 9
0 1 −10 0
−2 1 0 −6