Question

# 2 and # 3

2 -6 4 -4 0 -4 6 1. Define A = 8 01 . Determine, by hand, the LU factorization, of A. You may of course check your answer using appropriate technology tools. Then use your result to solve the system of equations Ax b, where b--4 2 0 5 2 2. Suppose A-6 -3 133Even though A is not square, it has an LU factorization A LU, 4 9 16 17 where L and U are 3-by-3 and 3-by-4 matrices, respectively. Determine the two matrices, L and U, showing all steps in the construction process. 3. Suppose a, b, c, and d are rea umbers and A- Show that A is invertible provided αメ0 and not two a, b, c, and d are equal. You may find it helpful to first determine the matrix U in the LU factorization of A. Then use the facts that the determinant of a triangular matrix is the product of its diagonal entries, the multiplicative property of the determinant (det (A) det(LU) det (L) det(U)), and the Invertible Matrix Theorem.

Answer 1

A = [2 6 4 0 -3 9 5 -13 16 2 3 17]_3 times 4 let A = LU here L_3 times 3 & V_3 times 4 so to get U, apply Gaussian elimination A = [2 6 4 0 -3 9 5 -13 16 2 3 17] R_2 rightarrow R_2 - 3R_1 R_3 rightarrow R_3 - 2R_1 [2 0 0 0 -3 9 5 -28 6 2 -3 13] R_3 rightarrow R_3 + 3R_2 U = [2 0 0 0 -3 0 5 -28 -78 2 -3 4] & L = [1 * * 0 1 * 0 0 1] corresponding use reverse row operations used in U i.e, R_2 rightarrow R_2 + 3R_1 R_3 rightarrow R_3 + 2R_1 R_3 rightarrow R_3 - 3R_2 L = [1 3 2 0 1 -3 0 0 1] \r\n A = [a a a a a b b b a b c c a b c d]_4 times 4 first