Question 6. Compute the LU factorization of each of the following matrices. 3 24 (a) (b)...

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Question 6. Compute the LU factorization of each of the following matrices. 3 24 (a) (b) -2 2 -2 01-01-19 (c) 4 Question 7. Find the LDU decomposition of the matrix 1 1 356 -2 27 Question 8. Let A be a 3 3 matrix and suppose that 2a1a24a3 = 0, -6 -3 4 where a1, a2, a3 are the column vectors of A. Does the linear system Ax = 0 have a non-trivial solution? Is A invertible? Question 9. Let U be an n x n upper triangular matrix with non-zero diagonal entries. Prove that U is invertible and U-1 is also an upper triangular matrix. Question 10. Show that if A is a symmetric nonsingular matrix, then A-1 is also sym- metric. Question 11. Perform each of the following block multiplications: 4 -21 1 1 1 -1 3 1 (a) 2 1 2 1 1 2 1 2 3 (b) 4 -2 1 2 12 0 (c) 00 0 0 0 10 0 00 Question 6. Compute the LU factorization of each of the following matrices. 3 24 (a) (b) -2 2 -2 01-01-19 (c) 4 Question 7. Find the LDU decomposition of the matrix 1 1 356 -2 27 Question 8. Let A be a 3 3 matrix and suppose that 2a1a24a3 = 0, -6 -3 4 where a1, a2, a3 are the column vectors of A. Does the linear system Ax = 0 have a non-trivial solution? Is A invertible? Question 9. Let U be an n x n upper triangular matrix with non-zero diagonal entries. Prove that U is invertible and U-1 is also an upper triangular matrix. Question 10. Show that if A is a symmetric nonsingular matrix, then A-1 is also sym- metric. Question 11. Perform each of the following block multiplications: 4 -21 1 1 1 -1 3 1 (a) 2 1 2 1 1 2 1 2 3 (b) 4 -2 1 2 12 0 (c) 00 0 0 0 10 0 00