Multiply both sides by 0

That doesn't help solve the problem. The reason why we would multiply both sides by a number is to remove denominators, but there is some math going on that people gloss over, because they do it in their head.

a
-  =  1
2

Multiply both sides by 2 to get 'a' alone.

a
-  * 2  = 1 * 2
2

Cancel the 2's and be done with it. But that is the shortcut and we really should write it as:

a     2
-  *  -   = 1 * 2
2     1

Multiplying both sides out gives us:

a * 2
------  = 2
   2

Now since multiplying fractions together involves multiplying the numerator and then the denominator to form a single fraction, we can reverse that process by factoring and take into account the communicative nature of multiplication to order them how we want before splitting them back into two individual fractions. Like so:

a * 2
------  = 2
1 * 2

a   2
- * -  = 2
1   2

Then the 2/2 reduces to 1 and it is pretty simple to reduce to a = 2. However, with your example, there are some complications.

0
-  =  a
0

0   0
- * - =  a * 0
0   1

So far, so good. Now, let's combine so that we can refactor... oh. That just results in the original equation. I mean the factors are still there and you could technically perform the steps from my example like so:

0   0
- * - =  a * 0
1   0

But, all you have managed to do is swap the sides where the 0/0 occurs. Without it being defined you can't really cancel out the 0 through multiplication.

TL;DR; Always show your work, kids.