Section 2.3: Using Matrices to Solve Systems with Three or More Unknowns

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A linear equation in three unknowns x, y and z is an equation of the form

where a, b, c, and d are numbers, and where a, b, and c are not all zero. (Two of them may be zero, or one of them, or none of them, but not all three of them.)

Setting Up a System of Linear Equations in Matrix Form

This is identical to what we learned in tutorial 2.2: the augmented matrix form of a single linear equation ax + by + cz = d is just the single row=matrix [abcd]. The augmented matrix of a whole system is then a matrix with one row for each equation in the system.

Using Row Reduction (Gauss-Jordan) to Solve Systems with Three Unknowns

Just as in the case of two unknowns, we shall use the following kinds of row operations:

The technique for reducing a matrix to row-reduced echelon form is the same as in Part C of the previous tutorial. (To go back there, press "prev. on the sidebar and then select Part C.) To illustrate this, we shall go through Example 2 on p. 140 of Finite Mathematics Applied to the Real World, or Finite Mathematics and Calculus Applied to the Real World. In that example, we solve the system

The augmented matrix for the system is

Now we go through the steps illustrated in Part B of the previous tutorial.

First Step: Clear all fractions and/or decimals (if any) by multiplying rows that contain them by a suitable number.

That step is not necessary here, as there are no fractions or decimals.

Second Step: Designate the first non-zero entry in the first row as the pivot.

Third Step: Use the pivot to clear its column using operations of type 3.

Simplification Step: If at any stage of the process, all the numbers in a row are multiples of an integer, divide by that integer -- a type 2 operation.

Fourth Step: Select the first non-zero entry in the next row as pivot, and go to Step 3.

Simplification Step:

Step 4:

At this point, we have run out of rows (there is no next row to go to) so we are done with the pivoting.

Final Step: Turn all the pivots into 1's by dividing each of the rows by the value of its pivot.

Now the matrix is reduced, so we translate back into equations to obtain the solution.

Consider the following sustem of equtions

The result of clearing fractions and decimals, and then performing the first pivot operation is:
1
1
0
4
0
0
4
1
16
4
0
2
2
2
1
0
0
1
2
1
1
1
0
4
0
0
2
1
8
2
0
1
1
1
1
0
0
1
2
1
1
1
0
4
0
0
4
1
16
4
0
2
2
2
1
0
0
1
2
1

The result of the next pivot operation (followed by a simplification step) is:
4
0
1
0
4
0
4
1
16
4
0
0
1
4
2
0
0
1
2
1
4
0
1
0
4
0
1
1

4
4
1
0
0
1
4
2
0
0
1
2
1
1
0
1
0
4
0
4
1
16
4
0
0
1
4
2
0
0
1
2
1

The result of the next two pivoting steps (including some simplification steps) is:
1
0
0
0
3
0
1
0
0
6
0
0
1
0
4
0
0
0
2
3
1
0
0
0
0
0
1
0
0
6
0
0
1
0
4
0
0
0
2
3
1
0
0
0
4
0
1
0
0
4
0
0
1
0
2
0
0
0
2
1

The solution of the system is:

Question
That is all very well. But what happens if the augmented matrix reduces to one of the non-standard forms, such as

Answer
Since row-reduced echelon form is as far as we can go with the matrix (go to Part C of the previous tutorial to find out more about row-reduced echelon form), we translate back into equations and obtain

Now we solve each of these equations for the first variable, obtaining

We can now choose z to be any number, and then get corresponding values for x and y according to the formulas, giving infinitely many solutions. Thus, the general solution is

We get particular solutions by choosing specific values for z. For example, z = 2 gives the particular solution

The following summary is adapted from p. 146 of Finite Mathematics Applied to the Real World and Finite Mathematics and Calculus Applied to the Real World. (Also, you can press the "summary" button on the sidebar to bring up a page with this and other information.)

Solutions of Systems of Linear Equations

One of three things will happen in any system of linear equations. There will be:

  • No Solutions This occurs when you end up with a row of the form 
     		[0   0   0  . . .  0   #] ,
    where # is a non zero number. As soon as you spot such a row, check to see that you haven't made an error and then stop. The given system has no solution, so there is no point in continuing any further.
  • Exactly One Solution (Unique Solution) This occurs if, at the conclusion of the row reduction, translation back into equations yields a single value for each of the unknowns.
  • Infinitely Many Solutions This occurs if, at the conclusion of the row reduction, translation back into equations does not yield a single value for each of the unknowns. In this case you can easily solve for the unknowns corresponding to the pivots, which will be the first unknowns in each equation. The other unknowns can be assigned arbitrary values. The arbitrary unknowns are called parameters, and the general solution as we have written it is called a parametrized solution. These are the only things that can happen. It is impossible, for example, to have a system of linear equations with exactly eleven solutions. If it has more than one solution, it must have infinitely many solutions.

Hints

  • If you end up with one or more rows of zeros, all this means is that one of the equations is redundant, and can thus be ignored.
  • If a system of linear equations has no solution, it is said to be inconsistent. The reason for this term is that the equations actually contradict one another. Here is an obvious example of such a contradiction. 
      	x + y + z = 0
	x + y + z = 44
    A reasonable person could hardly expect x, y, and z to add up to 0 and 44 at the same time!
  • A system of linear equations where there are fewer equations than unknowns is said to be underdetermined. These are the systems that usually give infinitely many solutions (as in Example 6) but may also result in no solutions if they are contradictory (as in the case immediately above). Such a system can never have a unique solution.
  • A system of equations in which the number of equations exceeds the number of unknowns is said to be overdetermined. In an overdetermined system, anything can happen, but such a system will usually be inconsistent.

Now try some of the exercises on pp. 147-150 of Finite Mathematics Applied to the Real World, or Finite Mathematics and Calculus Applied to the Real World.

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Last Updated: January, 1997
Copyright © 1997 StefanWaner and Steven R. Costenoble