## 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.