Unique Solution, No Solution, or Infinite Solutions

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Learning Objectives

By the end of this section you should be able to:

1. Understand the diffrence between unique solutions, no solutions, and infinitely many solutions.
2. Reconize when a matrix has a unique solutions, no solutions, or infinitely many solutions.
3. Reconize when a matrix has a unique solutions, no solutions, or infinitely many solutions using python.

Unique Solution

The example shown previously in this module had a unique solution. The structure of the row reduced matrix was

\[\begin{split}\begin{vmatrix} 1 & 1 & -1 & | & 5 \\ 0 & 1 & -5 & | & 8 \\ 0 & 0 & 1 & | & -1 \end{vmatrix}\end{split}\]

and the solution was

\[x = 1\]

\[y = 3\]

\[z = -1\]

As you can see, each variable in the matrix can have only one possible value, and this is how you know that this matrix has one unique solution

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No solution

Let’s suppose you have a system of linear equations that consist of:

\[x + y + z = 2\]

\[y - 3z = 1\]

\[2x + y + 5z = 0\]

The augmented matrix is

\[\begin{split}\begin{vmatrix} 1 & 1 & 1 & | & 2 \\ 0 & 1 & -3 & | & 1 \\ 2 & 1 & 5 & | & 0 \end{vmatrix}\end{split}\]

and the row reduced matrix is

\[\begin{split}\begin{vmatrix} 1 & 0 & 4 & | & 1 \\ 0 & 1 & -3 & | & 1 \\ 0 & 0 & 0 & | & -3 \end{vmatrix}\end{split}\]

As you can see, the final row states that

\[0x + 0y + 0z = -3\]

which impossible, 0 cannot equal -3. Therefore this system of linear equations has no solution.

Let’s use python and see what answer we get.

import numpy as py
from scipy.linalg import solve

A = [[1, 1, 1], [0, 1, -3], [2, 1, 5]]
b = [[2], [1], [0]]

x = solve(A,b)
x

---------------------------------------------------------------------------
LinAlgError                               Traceback (most recent call last)
<ipython-input-1-afc47691740d> in <module>()
      5 b = [[2], [1], [0]]
      6
----> 7 x = solve(A,b)
      8 x

C:\Users\Said Zaid-Alkailani\Anaconda3\lib\site-packages\scipy\linalg\basic.py in solve(a, b, sym_pos, lower, overwrite_a, overwrite_b, debug, check_finite, assume_a, transposed)
    217         return x
    218     elif 0 < info <= n:
--> 219         raise LinAlgError('Matrix is singular.')
    220     elif info > n:
    221         warnings.warn('scipy.linalg.solve\nIll-conditioned matrix detected.'

LinAlgError: Matrix is singular.

As you can see the code gives us an error suggesting there is no solution to the matrix.

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Infinite Solutions

Let’s suppose you have a system of linear equations that consist of:

\[-3x - 5y + 36z = 10\]

\[-x + 7z = 5\]

\[x + y - 10z = -4\]

The augmented matrix is

\[\begin{split}\begin{vmatrix} -3 & -5 & 36 & | & 10 \\ -1 & 0 & 7 & | & 5 \\ 1 & 1 & -10 & | & -4 \end{vmatrix}\end{split}\]

and the row reduced matrix is

\[\begin{split}\begin{vmatrix} 1 & 0 & -7 & | & -5 \\ 0 & 2 & -3 & | & 1 \\ 0 & 0 & 0 & | & 0 \end{vmatrix}\end{split}\]

As you can see, the final row of the row reduced matrix consists of 0. This means that for any value of Z, there will be a unique solution of x and y, therefore this system of linear equations has infinite solutions.

Let’s use python and see what answer we get.

import numpy as py
from scipy.linalg import solve

A = [[-3, -5, 36], [-1, 0, 7], [1, 1, -10]]
b = [[10], [5], [-4]]

x = solve(A,b)
x

C:\Users\Said Zaid-Alkailani\Anaconda3\lib\site-packages\scipy\linalg\basic.py:223: RuntimeWarning: scipy.linalg.solve
Ill-conditioned matrix detected. Result is not guaranteed to be accurate.
Reciprocal condition number: 3.808655316038273e-19
  ' condition number: {}'.format(rcond), RuntimeWarning)

array([[-12.],
       [ -2.],
       [ -1.]])

As you can see we get a different type of error from this code. It states that the matrix is ill-conditioned and that there is a RuntimeWarning. This means that the computer took to long to find a unique solution so it spat out a random answer. When RuntimeWarings occur, the matrix is likely to have infinite solutions.