Consider the matrixA =a. Find lower triangular elementary matrices E\, E2, ..., Em such

Chapter 2, Problem 90

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Consider the matrixA =a. Find lower triangular elementary matrices E\, E2, ..., Em such that the productEm E2E\Ais an upper triangular matrix U. Hint: Use elementary row operations to eliminate the entries below the diagonal of A. b. Find lower triangular elementary matrices M\, M2,..., Mm and an upper triangular matrix U such thatA = M\M2 MmU.c. Find a lower triangular matrix L and an upper triangular matrix U such thatA = LU.Such a representation of an invertible matrix is called an LU -factorization. The method outlined in this exercise to find an LU-factorization can be streamlined somewhat, but we have seen the major ideas. An LU- factorization (as introduced here) does not always exist (see Exercise 92).d. Find a lower triangular matrix L with ls on the diagonal, an upper triangular matrix U with ls on the diagonal, and a diagonal matrix D such that A = LDU. Such a representation of an invertible matrix is called an LDU-factorization.