Question

Let A= [ 2 4 ] and b= [ 2 ]

[ 8 17 ]. [ 11 ]

(a) Factor A = LU

b) Use the factorization from part (a) to solve Ax = b.

Answer #1

Solve the equation Ax = b by using the LU factorization given
for A.
A= [4 -6
4
= [1 0 0 [4 -6
4 [0
-12 15
-7
-3 1 0 0 -3
5 b= 12
12 -15
8]
3 -1 1] 0 0
1] -12]
Let Ly=b and Ux=y. Solve for x and y.
y=
x=

Find LU factorization of A.solve the system Ax=b using LU
fact
A=(1 -3 0, 0 1 -3,2 -10 2). b=(-5,-1,-20)

A=
2
-3
1
2
0
-1
1
4
5
Find the inverse of A using the method: [A | I ] → [ I | A-1 ].
Set up and then use a calculator (recommended). Express the
elements of A-1 as fractions if they are not already integers. (Use
Math -> Frac if needed.) (8 points)
Begin the LU factorization of A by determining a first
elementary matrix E1 and its inverse E1-1. Identify the associated
row operation. (That...

Let A = [1 5 ; 3 1 ; 2 -4] and b = [1 ; 0 ; 3] (where semicolons
represent a new row)
Is equation Ax=b consistent?
Let b(hat) be the orthogonal projection of b onto Col(A). Find
b(hat).
Let x(hat) the least square solution of Ax=b. Use the formula
x(hat) = (A^(T)A)^(−1) A^(T)b to compute x(hat). (A^(T) is A
transpose)
Verify that x(hat) is the solution of Ax=b(hat).

Problem 8 Let a(t) =/= b(t) be given. The factorization for a
second order ODE is commutative if (D + a(t) I) (D + b(t) I) y = (D
+ b(t) I) (D + a(t) I) y.
• Find condition on a(t) and b(t) so that the factorization is
commutative.
• Find the fundamental set of solutions for a second order ODE
that has a commutative factorization.
• Use the above results to find the fundamental set of solutions
of...

Given:
A =
1
−1
−2
3
B =
1
5
8
−8
C =
1
2
3
4
Solve:
AX + B = C
X =

Find the LU factorization of the matrix
a)
( 6 2 0
-12 -3 -3
-6 -1 -14
9 -12 45 )
b) ( 2 -1 -2 4
6 -8 -7 12
4 -22 -8 14 )

Problem 3.4.43 Let f(x) = x^4 - 2x^3 + x^2 + 12x + 8.
(a) List possible rational zeros.
(b) Use synthetic division to identify a zero and then factor the
polynomial completely.
(c) Identify each zero and the multiplicity of each zero

Let A = {0, 3, 6, 9, 12}, B = {−2, 0, 2, 4, 6, 8, 10, 12}, and C
= {4, 5, 6, 7, 8, 9, 10}.
Determine the following sets:
i. (A ∩ B) − C
ii. (A − B) ⋃ (B − C)

Let A =
v1
v2
v3
v1
2
8
16
v2
8
0
4
v3
16
4
1
be the ADJACENCY MATRIX for an undirected graph G. Solve the
following:
1) Determine the number of edges of G
2) Determine the total degree of G
3) Determine the degree of each vertex of G
4) Determine the number of different walks of length 2 from
vertex v3 to v1
5) Does G have an Euler circuit? Explain

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