If A and B are two matrices such that AB=A+B then det (A-I3)=?

1 Answer
Jul 12, 2018

[Assuming A,B are 3x3 (square) matrices]

I don't know what form you are looking for but these may provide some help.

  • From the given condition:

#AB = A + underbrace(B)_(= bbbI B) #

# (A - bbbI)B = A #

Assuming required invertibility, ie #det M ne 0#

# det (A - bbbI) det B = det A #

  • #bb( det(A - bbbI )= (det A)/ (det B) )#

  • Re-jigging the condition:

#(A - bbbI)(B- bbbI) = underbrace(AB - A - B)_( := 0) + bbbI#

Again assuming required invertibility, ie #det M ne 0#

  • # bb( det (A - bbbI) = 1/(det (B- bbbI)) )#

PS: You can also use the initial condition to show that # [A,B] = 0 #, but don't immediately see a way to use that so as to get a cute solution.