By factorization, we noted in the LU Decomposition method of solving linear systems(i.e., typically Ax = b) that A = LU where Ly= b and UX = y to solve the system. Given thematrices A and U, what are the values of a and u? Assume that the matrix will have aconsistent and unique solution (no other assumptions are made). Show all solutions andexplanations.A =3113136113U= 300118/3и0 5/2

The given matrix is A=3111311a3. Let A=LU where L is lower triangular matrix and U is upper…

Answered: By factorization, we noted in the LU…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Consider the following system of linear equations (SLE) 2x +y = 7 3x +2y = 12 Solve this SLE geometrically by plotting the two equations on a graph. Compose the augmented matrix corresponding to this SLE. By transforming the augmented matrix to form an echelon form matrix, solve the SLE. Detail each of the transformation steps.
Determine whether the statement below is true or false. Justify the answer. The equation Ax = b is homogeneous if the zero vector is a solution. Choose the correct answer below. OA. The statement is false. A system of linear equations is said to be homogeneous if it can be written in the form Ax=b, where A is an mxn matrix and b is a nonzero vector in R™. Thus, the zero vector is never a solution of a homogeneous system. B. The statement is true. A system of linear equations is said to be homogeneous if it can be written in the form Ax=b, where A is an mxn matrix and b is a nonzero vector in R™. If the zero vector is a solution, then b = 0. C. The statement is true. A system of linear equations is said to be homogeneous if it can be written in the form Ax = 0, where A is an mxn matrix and 0 is the zero vector in R™. If the zero vector is a solution, then b = Ax = A0 = 0. O D. The statement is false. A system of linear equations is said to be homogeneous if it can be written in the form…
Use a graphing utility or computer software program with matrix capabilities to write v as a linear combination of u1, U2, U3, U4, Ug and ug. Then verify your solution. u1 = (1, 1, -4, -4, 3, 4) u2 = (4, -3, 1, -1, 1, 4) u3 = (0, 4, -3, -2, -4, 1) U4 = (4, 3, -4, 3, 4, 3) U5 = (4, -2, -4, 2, -4, 4) U6 = (3, 4, -4, 3, 3, 4) v = (-6, 42, -20, -18, 51, 11) V =
3. Consider the system of linear equations 2x₁ + 3x₂ + x3 = −1 3x₁ + 3x₂ + x3 = 1 3x₁ + 4x₂ + x3 = -2 a. Find the inverse of the coefficient matrix A. b. Using the inverse matrix of A, find the solution of the system. 4. Suppose the following matrix A is the augmented matrix for a system of linear equations A = 1 2 3 4 2 -1 -2 a² -1 -7 -11 a where a is a real number. Determine all the values of a so that the corresponding system is consistent.
Use a software program or a graphing utility with matrix capabilities to write v as a linear combination of u1, u2, u3, u4, and u5. Then verify your solution. (Enter your answer in terms of u1, u2, u3, u4, and u5.)v = (6, 1, −10, 11, 8 )u1 = (1, 2, −3, 4, −1)u2 = (1, 2, 0, 2, 1)u3 = (0, 1, 1, 1, −4)u4 = (2, 1, −1, 2, 1)u5 = (0, 2, 2, −1, −1)
Ques.5 Find the inverse of the matrix involve in the system by the Gauss-Jordan method and then find the solution X + y + z = 1 4x + 3y -z = 6 3x + 5y + 3z = 4
I. Consider the following linear system: 3x1 + 9x2 + 3x3 = 6 2x1 — 2х1 — 4x2 X3 = 12 = -8 We can view this system as a matrix equation: 3 9 31 0 -1 2 12 -2 -4 Which has the form: Ax X = B To solve the original system, we can solve this equation for X by multiplying by A-1: A-1x (A x X) = A-! × B (A-1 × A) × X = A-1 × B Iz x X = A-1 × B X = A-1 x B Use the bottom equation to solve for X. Find A-1 for the matrix A and solve the original system by using A-1 and Bin the (1) equation below. X = A-1 x B (2] Use the matrix A² you just found to solve the system: 3x1 + 9x2 + 3x3 = 12 2x1 -2x1 – 4x2 X3 = 9 = -3
Let A be a 3x3 matrix, and let u₁ and u₂ be the first two columns of A. Assume that u₁ and u₂ are not parallel. If a linear system Av = b has infinitely many solutions, what can we say about the third column u3 of A, in terms of u₁ and u₂? In other words, how does u3 relate to u₁ and u₂?
Let A and B be matrices of thesame size. If x is a solution to both the system Ax =0 and the system Bx = 0, show that x is a solution tothe system (A + B)x = 0. Provide reasoning/details please.
a. Construct a 3 x 2 matrix A such that A = 0 has only the trivial solution. A = b. Construct a 3 x 2 matrix B such that Ba B = = O has a nontrivial solution.

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