Ill Conditioned Systems of Linear Simultaneous Equations

A system of linear simultaneous equations is ill conditioned if a small change in any of the coefficients results in a big change in the solutions to the system.
Consider the system of equations
$2x_1+x_2=5$

$1.99x_1+x_2=3$

The equations are the same except that the coefficient of
$x_1$
in the first equation is less by 0.01.
Subtracting the second equation from the first gives
$0.01x_1=2 \rightarrow x_1=2/0.01=200$

Then from the first equation
$2 \times 200+x_2=5 \rightarrow x_2=5-400=-395$
.
Now suppose that the coefficient of
$x_1$
in the second equation is changed from 1.99 to 2.01. The system of equations becomes
$2x_1+x_2=5$

$2.01x_1+x_2=3$

Subtracting the second equation from the first gives
$-0.01x_1=2 \rightarrow x_1=2/-0.01=-200$

Then from the first equation
$2 \times -200 +x_2=5 \rightarrow x_2=2 \times 200+5=405$

Changing one coefficient by 0.02 or 1% have changed the solutions for
$x_1, \: x_2$
by large amounts.
Systems of linear equations are ill conditioned in general the equations are nearly linearly dependent or if the coefficient matrix has a small determinant.
$\left| \begin{array}{cc} 2 & 1 \\ 1.99 & 1 \end{array} \right| =2 \times 1 - 1 \times 1.99 =0.01$

The determinant of the coefficient matrix is nearly zero so the system is ill conditioned. 