Problem of the week - Classify the system of linear equations

Systems of linear equations lie at the heart of linear algebra, and they are used to solve practical problems in many fields of study. We can find examples in biology, economics and electronics, to cite only a few examples, where the solution to several problems is reduced to solve a system of linear equations.

So, this week, the problem will be related to this topic, where you should prove you understand the basic concepts when finding the solution of a system of linear equations.

Given the following system of linear equations:

-2x +  y +  z = -1
 6x - 2y - 4z = 2a
-5x + 2y + az = -4

Which values of 'a' lead to each of these possibilities?

1. The system is inconsistent

2. The system is consistent dependent

3. The system is consistent independent

This problem can be solved by different methods, but here I will do it by using Gaussian elimination on the augmented matrix.

The augmented matrix for this system is:

┌                  ┐
│ -2   1   1 |  -1 │
│  6  -2  -4 |  2a │
│ -5   2   a |  -4 │
└                  ┘

Perform row operations to convert the augmented matrix to row echelon form.

f3 <———> -2•f3
┌                   ┐
│ -2   1    1 |  -1 │
│  6  -2   -4 |  2a │
│ 10  -4  -2a |   8 │
└                   ┘
f2 <———> f2 + 3•f1
f3 <———> f3 + 5•f1
┌                      ┐
│ -2  1      1 |    -1 │
│  0  1     -1 |  2a-3 │
│  0  1  -2a+5 |     3 │
└                      ┘
f3 <———> f3 - f2
┌                       ┐
│ -2  1      1 |     -1 │
│  0  1     -1 |   2a-3 │
│  0  0  -2a+6 |  -2a+6 │
└                       ┘

The system of linear equations if consistent if and only if the rank of the coefficient matrix is equal to that of the augmented matrix.

The rank of the coefficient matrix could be equal to 2 or 3 depending on the expression -2a+6. When the expression is zero the rank is 2, otherwise is 3.

The expression is equal to 0 when a = 3, but notice that for this value the rank of the augmented matrix is also equal to 2, so the system is always consistent.

So:

1 - None of the values of 'a' results in an inconsistent system of linear equations.

2 - When a=3 the rank is equal to 2 and system is consistent dependent (the rank is less than the number of unknowns)

3 - When a<>3 the rank is equal to 3 and the system is consistent independent (the rank is equal to the number on unknowns)