Question

Another linear algebra question...
Proof of: If A and B are nonsingular matrices, (A+B) is not necessarily invertible.

Answer 1

I'm just assuming A = I and B=-I. Is there a generalization for this?

Answer 2

you can do this proof by example
think up two matrices whose determinant is not zero, but the sum has determinant zero
examples abound

Answer 3

Well for proving negation if you can provide even one case then you are good to go

Answer 4

I used an example to prove. But is there any formal way to describe this proof?

Answer 5

Well you can if you want...

Answer 6

Let's assume that for every non singular and invertible A and B, A+B is non singular and invertible...

Then that means...

\(|A| \ne 0\) and \(|B| \ne 0\)

Now, \(|A+B| = |A| + |B| \ne 0 \iff |A| \ne -|B|\)

But this is possible... you can here provide the example of A = I and B = -I

Answer 7

I thought as much... All the proofs I'm coming along are long but this one seems simple enough.

Answer 8

Using \(det(A)\) method is out of scope for now, though

Answer 9

Yeah

Answer 10

Seems I can't help but use it xD. All right, thanks for the help again!

Answer 11

:) Glad to be of help

Answer 12

Any other proofs?

Answer 13

Not using determinants, I mean. For now it's out of scope;

Answer 14

I was only gonna say this, but it's so silly I decided not to. Oh well here it is haha

If A is nonsingular, B=-A is nonsingular but clearly A+B = A-A = 0 is singular. So it's a mild generalization of what you were asking but nothing too special really haha.

Answer 15

All right, fair enough xD

Answer 16

So a general approach is A+B=0. Not too much mathematical formalism but I'm OK with this XD

Answer 17

There's probably better answers out there, but this is pretty good to me haha.

Answer 18

All right, thanks!