If a matrix is singular then it cannot have an $LU$ factorization. True or False
I have to show: (a) non-singularity, (b) representation, (c) inverse and (d) union
LU decomposition and singular matrix
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matrices
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0False, of course: the zero matrix $O$ has $L=I$ (identity) and $U=O$. What the following part means, I can't understand. – 2017-02-21
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0This is what I'm given as "hint": Following the definition of an elementary matrix M_k = I -m_k e_k^T , you can split this into an identity matrix additively combined with a matrix obtained from m_k e_k^T (outer product of a column vector and row vector). In order to prove the inverse property, try a straightforward multiplication of (I-m_k e_k^T) with (I+m_k e_k^T) and remorselessly (?) carry out the multiplication term by term – 2017-02-21