Matrices -- I think it's a false statement.

[TheWrathOfMath]

Is the set of all matrices such that AB-A=In a linear subspace of V=F^nxn?
A exists in V and B is a constant matrix of order nxn.

I think it is false, but I do not know how to disprove it.

 

[Steven G]

You disprove it by finding a counterexample. You need try this on your own, especially if you want help from this forum.

 

[TheWrathOfMath]

You disprove it by finding a counterexample. You need try this on your own, especially if you want help from this forum.

AB−A=I
CB−C=I
AB−A+CB−C=2I⇒(A+C)B−(A+C)=2I

What about a numerical example, though?

 

[Steven G]

AB−A=I
CB−C=I
AB−A+CB−C=2I⇒(A+C)B−(A+C)=2I

What about a numerical example, though?

...what about a numerical example?
I am starting to think that you are not understanding what is meant by a math help forum. For the record, it is a forum where we help students solve their problems. What we never do is solve problems for students.
You need to dig deeper and understand what is going on. Someone giving you answers will just prevent you from doing that.

 

[topsquark]

AB−A=I
CB−C=I
AB−A+CB−C=2I⇒(A+C)B−(A+C)=2I

What about a numerical example, though?

First, is this a condition on A that it's in a linear subspace or is it a condition on B that it's in a linear subspace? (Or both?)

Pick a matrix A. Then solve for B. What does that mean for A or B (or both) to be in the subspace?

-Dan

 

[TheWrathOfMath]

First, is this a condition on A that it's in a linear subspace or is it a condition on B that it's in a linear subspace? (Or both?)

Pick a matrix A. Then solve for B. What does that mean for A or B (or both) to be in the subspace?

-Dan

I managed to solve it. Thank you