It's impossible to send them to an invalid address, BUT it's entirely possible to send them to an address for which no one has the key.
Take for example: 1BitcoinEaterAddressDontSendf59kuE
Check it out on blockchain. If you can brute force the private key, the coins are yours. Is it impossible? Theoretically, no, but practically...
Let's say you had a super computer that was guessing 999 trillion keys per second. It would take you 3.5 billion years to exhaust just 10% of the keyspace, which means in 3.5 billion years you would have a 10% chance of having guessed the key. Good luck with those odds!
Actually - given enough time - is it theoretically possible to crack the private key to that address?
I mean in the future computers will be 1000x more powerful than they are today.
Will our brains be blown out of our bodies?
See Syke's answer.
Also:
private key is a 256 bit integer.
public key is a pair of 256 bit integers giving the (x,y) coordinates of a point,
or a single 256 bit x coordinate and a parity bit used to reconstruct y.
address is a hash of the public key.
The bitcoin eater was made at step 3. Because it was made at step 3, we don't know if there are any points on our curve that can be hashed to give that address. This point isn't well understood around here.