1.Determine whether each of the following statements is True or False. (Nojustification necessary.)(a) If a matrix equation Ax = b has more than one solution, then the matrix equation Ax = 0 must alsohave more than one solution.(b) If A is a 3 x 3 matrix and the matrix equation Axhas a unique solution, then A must beinvertible.(c) If A is a 2 x 2 matrix, det A = 2, and B = A², then det 2B = 8.(d) Row operations on a matrix can change its null space.(e) If A is a diagonalizable matrix, then the columns of A must be linearly independent.

As per bartleby guideline for more than three sub parts asked only first three are to be answered…

1. Determine whether each of the following statements is True or False. (No justification necessary.) (a) If a matrix equation Ax = b has more than one solution, then the matrix equation Ax = 0 must also have more than one solution. (b) If A is a 3 x 3 matrix and the matrix equation Ax = 0 has a unique solution, then A must be invertible. (c) If A is a 2 x 2 matrix, det A = 2, and B = A², then det 2B = 8. (d) Row operations on a matrix can change its null space. (e) If A is a diagonalizable matrix, then the columns of A must be linearly independent.

1. Determine whether each of the following statements is True or False. (No justification necessary.) (a) If a matrix equation Ax = b has more than one solution, then the matrix equation Ax = 0 must also have more than one solution. (b) If A is a 3 x 3 matrix and the matrix equation Ax = 0 has a unique solution, then A must be invertible. (c) If A is a 2 x 2 matrix, det A = 2, and B = A², then det 2B = 8. (d) Row operations on a matrix can change its null space. (e) If A is a diagonalizable matrix, then the columns of A must be linearly independent.

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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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

1.
Determine whether each of the following statements is True or False. (No
justification necessary.)
(a) If a matrix equation Ax = b has more than one solution, then the matrix equation Ax = 0 must also
have more than one solution.
(b) If A is a 3 x 3 matrix and the matrix equation Ax
has a unique solution, then A must be
invertible.
(c) If A is a 2 x 2 matrix, det A = 2, and B = A², then det 2B = 8.
(d) Row operations on a matrix can change its null space.
(e) If A is a diagonalizable matrix, then the columns of A must be linearly independent.

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