Question

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

Answer 1

[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.