Singular MatricesA matrix is singular if its inverse does not exist. One way to test for singularity is to compute a determinant; this will be zero if the matrix is singular. For example, the following matrix is singular. In[1]:=  |
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For many right-hand sides there is no solution. In[3]:=  |
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However, for certain values there will be a solution. In[5]:=  |
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For the first example, the rank of  does not equal that of the augmented matrix. This confirms that the system is not consistent and cannot be solved by LinearSolve. In[7]:=  |
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For the second example, the rank of  does equal that of the augmented matrix. This confirms that the system is consistent and can be solved by LinearSolve. In[8]:=  |
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In those cases where a solution cannot be found, it is still possible to find a solution that makes a best-fit to the problem. One important class of best-fit solutions involves least squares solutions and is discussed in a later section.
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