Question

# Suppose a nonhomogeneous system of six linear equations in eight unknowns has a solution, with two free variables. Is it possible to change some constants on the equations’ right sides to make the new system inconsistent?

Equations
Suppose a nonhomogeneous system of six linear equations in eight unknowns has a solution, with two free variables. Is it possible to change some constants on the equations’ right sides to make the new system inconsistent?

2020-12-16

Step 1
According to the given information, a non-homogeneous system of six linear equations in eight unknown has a solution with two free variables.
It is required to check whether is it possible to change some constants on the equations on right side to make new system inconsistence.
$$\displaystyle\dim{\left({N}\underline{{{A}}}\right)}$$ is same as number of free variables
so, $$\displaystyle\dim{\left({N}\underline{{{A}}}\right)}={2}$$
Step 2
Since, n = 8 as system has eight unknowns so, rank of coefficient matrix is:
rank(A)=8-2=6
Step 3
Further, col(A) is a six-dimensional subspace of $$\displaystyle{R}^{{6}}$$ because it has six linear equations so, $$\displaystyle{c}{o}{l}{\left({A}\right)}={R}^{{6}}$$
This means that Ax = b has a solution for every b and cannot change constant on right side to make system inconsistence.
Hence, it is not possible to change the constants.