Inverse of a Singular Matrix

For the inverse of a matrix A the following properties hold AA = I and A A = 1. Using the Gauss-Jordan ehmination method, one can easily calculate the inverse of a matrix. It can be shown that 

In some cases the inverse of the matrix cannot be calculated since the matrix becomes ill-conditioned, then one could calculate the so-called pseudo-inverse, which will be explained in the following sectioa 

When the colmtms of a matrix A are not hnearly independent, the inverse of the matrix can not be found, since there are many solutions to Ay=b. In this case it is still possible, however, to find a least squares solution to the problem. The matrix that solves this problem is called pseudo-inverse it can be calculated by using singular value decomposition. 

In case the columns of matrix are hnearly dependent (A is collinear), one can use singular value decomposition to calculate the pseudo inverse. A matrix (w, ) is factored into  

This matrix is singular since the third column is linearly dependent on the other two, it is equal to the first column plus two times the second colunrn. If one uses Matlab and one enters = [2 1 4 2 2 6 -4 2 0] 5 = im(A), then Matlab returns that the matrix is singular and the result is undetermined. 

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