4. A matrix A has a LU factorization if A = LU where L is a lower triangular matrix (a matrix with zeros abovediagonal) and U is an upper triangular matrix (a matrix with zeros below the diagonal).(a) Verify that A =4-483 -5-5 76 -8Is a product of a L =0-1 1 0 and U =2 0(b) Use the LU factorization to solve Az =H(c) Find L-¹ and U-¹, then use these to find A showing that A is invertible.43 -50-2 20 0 2for . To do so, first solve Lij= 6, and after Uz = ý.(d) (Optional) To find the LU factorization, reduce A to U by using only row replacements operations. Second,place entries in L so that the same sequence of row operations reduces L to I. Follow this algorithm to find theLU decomposition of A. (Consult Section 2.5 for more details.)

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4. A matrix A has a LU factorization if A = LU where L is a lower triangular matrix (a matrix with zeros above diagonal) and U is an upper triangular matrix (a matrix with zeros below the diagonal). (a) Verify that A = 4 -4 8 3 -5 7 -8 Is a product of a L = 00 -1 1 2 0 1 and U = 4 3 -5 0 -2 2 0 0 2 (b) Use the LU factorization to solve Az = -4 for . To do so, first solve Ly=b, and after Uz = y. 6 (c) Find L¹ and U-1, then use these to find A showing that A is invertible.