Matrices - LU Decompostion

Question

Is LU Decomposition possible for every matrix ?

Answer 1

NO , every matrix doesnt have an LU decomposition

An invertible matrix has LU decomposition only if all its leading submatrices have non zero determinants

For example

If u have a 3 X 3 matrix

1 X 1 should have determinant non zero

all possible 2X2 from that should have  determinant non zero

and finally the 3 X 3 matrix also should have  determinant non zero

Then LU decomposition exists

Comments

@akash

because inly if  determinant not zero it will be invertible

so the matrix should be invertible and in addition to that the submatrices also should have determinant noot equal to zero

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@aish here it is given LU decomposition of singular matrices is not unique

https://math.stackexchange.com/questions/445457/show-that-the-lu-decomposition-of-matrices-of-the-form-beginbmatrix0-x-0

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I think it is ambiguous since its not unique

so cannot find a proper decomposition