Q25E State true or false, there exist... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 4: Q25E (page 200)

State true or false, there exist $2 脳 2$ matrix $A$ such that the space $V$ of all matrices commuting with $A$ is two-dimensional.

Short Answer

Expert verified

Therefore, the statement is True.

Step by step solution

Determine the set.

Consider a $2 脳 2$ matrix A from the space V that commute with all the element of S where $S = {B 鈭� M^{2 脳 2} AB = BA}$ .

If A=[0], determine the set S .

If $A = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$ then for any $B 鈭� M^{2 脳 2}$ the matrix A commute B with implies $S = M^{2 脳 2}$ .

Therefore, the dimension of the set $S$ is $4$ if $A = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$ .

If A=[1001], determine the set S.

If $A = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$ then for any $B 鈭� M^{2 脳 2}$ the matrix $A$ commute with $B$ implies $S = M^{2 脳 2}$ .

Therefore, the dimension of the set $S$ is if $A = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$ .

If A is linearly independent with l , determine the setS.

If $A$ is linearly independent with respect to $l$ then the set $S$ contain at-least two elements $A$ and $l$ .

Therefore, the dimension of the set $S$ is at least $2$ if is linearly independent with respect to $l$ means there exist $2 脳 2$ matrix $A$ such that the space $V$ of all matrices commuting with $A$ is at least two-dimensional.

Hence, the statement is true.

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91影视

State true or false, there exist $2 脳 2$ matrix $A$ such that the space $V$ of all matrices commuting with $A$ is two-dimensional.

Short Answer

Step by step solution

Determine the set.

If A=[0], determine the set S .

If A=[1001], determine the set S.

If A is linearly independent with l , determine the setS.

One App. One Place for Learning.

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91影视

State true or false, there exist 2脳2 matrixA such that the spaceV of all matrices commuting withA is two-dimensional.

Short Answer

Step by step solution

Determine the set.

If A=[0], determine the set S .

If A=[1001], determine the set S.

If A is linearly independent with l , determine the setS.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Statistics

Decision Maths

Logic and Functions

Calculus

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

State true or false, there exist $2 脳 2$ matrix $A$ such that the space $V$ of all matrices commuting with $A$ is two-dimensional.