There was only one way to find out.
On his way to Hyrule Castle, Link stopped for a moment to rest after climbing up some vines to reach a ledge. After resting for a few moments, he heard the sound of someone else climbing up the vines. As he peaked over to see who it was...
...he saw himself.
The climbing Link was so shocked by the sight of his face appearing right in front of him that he lost his grip on the vines. He fell backwards, his body twisting in midair, and he landed on his head. The impact caused his head to twist sharply to the side, snapping his neck and instantly killing him.
Aside from extreme shock and surprise, the other Link felt nothing but a sharp pain in his neck as he collapsed and died. His limp body went tumbling off the cliff and lay next to his double's body.
Suppose you traveled back in time and killed yourself. As a result, you would never have been alive to kill yourself, so your past self would still be alive.
The Link who had been climbing slowly began to stir. With a groan, he rose to his feet, and rubbed his neck, which felt extremely sore. He looked down with amazement at a corpse lying before him, which looked exactly like him.
And, if you were still alive, you would eventually travel back in time and kill yourself.
The Link that had been resting rose to his feet, and watched as his hand involuntarily grasped his sword's hilt, drew it from its sheath, and plunged it into the neck of a boy who looked exactly like him. Then a stab wound suddenly appeared in his neck, and as blood gushed out, both boys collapsed.
As a result, you would never have been alive to kill yourself, so your past self would still be alive.
The climbing Link rose up.
And, if you were still alive, you would eventually travel back in time and kill yourself.
The resting Link came back to life and stabbed his past self again.
And so on.
The two boys continued to die, come back to life, and die again.
Thus, each possibility seems to imply its own negation, a type of logical paradox.
However, since the mechanics of the universe are governed by probabilities, any unmeasured entity (in this case, your historical self) has numerous probable states. When that entity is measured, the number of its probable states singularizes, resulting in a single outcome (in this case, ultimately, you). Therefore, since the outcome of your past self is known, you killing your past self would be incompatible with that outcome. Thus, the outcome of one's trip backwards in time must be complementary with the state from which one left.
Both Links instantly transformed into zombies that had been dead all along and could survive broken necks and stab wounds.
The Multiple Timelines theory suggests that every seemingly random event with a non-zero probability actually occurs in all possible ways in different "Timelines", so that history is constantly branching into different alternatives. If backwards time travel is possible, it should result in the traveler ending up in a different branch of history than the one he departed from.
Link blinked and opened his eyes to find himself alone, with no other Link in sight. He looked to his left, and saw someone approaching. He quickly hid around a corner. When he heard the sound of someone climbing vines, he looked around the corner and up to see himself climbing up the vines. He sighed with relief, and quickly tried to leave the area - but a fairy suddenly flew into his path.
"Hey, look! This kid looks just like you, Link!"
The climbing Link turned around to see who Navi was indicating, and was so shocked that he lost his grip on the vines.