Pathological Systems

• When a real physical situation is modeled 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, at least as far as the equations 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, at least in principle.
• Consequently, it must always be theoretically possible to avoid divisions by zero when the set of equations has a solution.
• Here is an example of a matrix that has no inverse: What is the LU equivalent of
  $$A=\left[ \begin{array}{rrr} 1 & -2 & 3 \\ 2 & 4 &-1 \\ -1 &-14 & 11\\ \end{array} \right]$$

  >> lu(A)
  ans =
   2.0000      4.0000     -1.0000
  -0.5000    -12.0000     10.5000
   0.5000      0.3333      0
  

  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.