2. In the linear system Ax = b,A = 1 2 4 2 6 0b = 27 −4(a) Show that x = 1 −1 7 be a solution of the linear system.(b) Show that b can be expressed as a linear combination of columns of A with scalars 1, −1,and 7.(c) Find XT A if it is defined. When x is not a solution of ATX= b? 1. Consider a set of linear equations:xi + 4x2 – 3x3 = 7x2 + 6x3 = 2kaz = 1%3Dwhere k is an arbitrary constant.(a) Write the system in matrix-vector form and form an Augmented matrix.(b) Is the linear system solvable for k = 0, and whey?(c) When does the system have unique solution? And find a solution considering k = 2.2. In the linear system Ar = b,[1 2A =2 6 027(a) Show that r =-1 be a solution of the linear system.7(b) Show that b can be expressed as a linear combination of columns of A with scalars 1,-1,and 7.(c) Find x"A if it is defined. Whey x is not a solution of A"x = b?3. (a) Considering an 3 x 3 Suppose, invertible and symmetric matrix A, prove that A-1 issymmetric.[1 2(b) For a matrix A =2 1show that AAT and AT A are symmetric.

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Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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2. In the linear system Ax = b,
A = 1 2 4

2 6 0

b = 27

−4

(a) Show that x = 1

−1

7
be a solution of the linear system.

(b) Show that b can be expressed as a linear combination of columns of A with scalars 1, −1,
and 7.
(c) Find X^T A if it is defined. When x is not a solution of A^TX= b?

1. Consider a set of linear equations:
xi + 4x2 – 3x3 = 7
x2 + 6x3 = 2
kaz = 1
%3D
where k is an arbitrary constant.
(a) Write the system in matrix-vector form and form an Augmented matrix.
(b) Is the linear system solvable for k = 0, and whey?
(c) When does the system have unique solution? And find a solution considering k = 2.
2. In the linear system Ar = b,
[1 2
A =
2 6 0
27
(a) Show that r =
-1 be a solution of the linear system.
7
(b) Show that b can be expressed as a linear combination of columns of A with scalars 1,-1,
and 7.
(c) Find x"A if it is defined. Whey x is not a solution of A"x = b?
3. (a) Considering an 3 x 3 Suppose, invertible and symmetric matrix A, prove that A-1 is
symmetric.
[1 2
(b) For a matrix A =
2 1
show that AAT and AT A are symmetric.

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