Properties of Invertible Matrices: A2-AB+BA-B2 is Singular

[abonaser]

Show the following:

If A and B are n x n matrices such that A - B is singular then A² - AB + BA - B² is also singular.


I really have no clue how to solve this, but I am guessing that AB does not equal BA, I don't know how that can help or be relevant but just in case


Thanks alot, any help is appreciated!

 

[micromass]

A²-AB+AB-B²=(A-B)(A+B).

Try computing the determinant of this matrix.

 

[abonaser]

alright I am not sure how to calculate the determinant, because we are not actually given any information besides to the matrices A and B are nxn matrices

 

[micromass]

Agreed, but the only thing you need to know is det(A-B)=0. Can you now calculate det((A-B)(A+B))?

 

[abonaser]

well if det(A-B)=0. Then det((A-B)(A+B))=0 and the matrix is singular!

I am trying to understand though, how did you know that det(A-B)=0?

and thanks a lot for all the help!

 

[micromass]

Well, you said that A-B was singular. Thus that means that det(A-B)=0...

 

[abonaser]

aha I see right, I am really lost with this whole matrices situation haha thanks a lot for the help!