Pathological Systems

• When a real physical situation is modelled by a set of linear equations, we can anticipate that the set of equations will have a solution that matches the values of the quantities in the physical problem (the equations should truly do represent it).
• Because of round-off errors, the solution vector that is calculated may imperfectly predict the physical quantity, but there is assurance that a solution exists.
• Here is an example of a matrix that has no inverse:
  $$A=\left[ \begin{array}{rrr} 1 & -2 & 3 \\ 2 & 4 &-1 \\ -1 &-14 & 11\\ \end{array} \right]$$

  
  Element A(3,3) cannot be used as a divisor in the back-substitution. That means that we cannot solve.
• The definition of a singular matrix is a matrix that does not have an inverse.