Tuesday, 2 December 2008

Meaning of AB = 0 and AB = 1

AB = 0.

The mathematical statement seems simple.

It means the multiplication of variable A and B equals zero.

Though it seems simple and direct, mistake in understanding the implication of the zero exists.

When we say AB = 0, we indirectly (and logically) deduce that A = 0 or B = 0.

This deduction is with taken regardless of what the other variable is.

When we say A = 0, B can be anything since 0 multiply "anything" = 0.
This is true vice versa for B = 0.

But what if AB = 1? or AB = x?

This is where misconception of the "logically" deduction happens.

Many maths learners assumed that since AB = 0 indicated A = 0 or B = 0,
AB=1 indicated A = 1 or B = 1 also!

This is a grave and serious mistake made.

AB= 1 does not imply A = 1 or B = 1 .

If A = "1" is true, then AB = 1 means "1" x B = 1, which is definitely false, as 1 x B = B!

Likewise when B= 1 is assumed.

Therefore the AB = 0 cannot be applied across the board to cover all else with the same deduction.

The equal to Zero has special meaning, and should not be confused with other number equated.

In summary: AB = 0 means A= 0 or B= 0 only if "= 0".

A little accurate understanding goes a long way.... in maths, especially.



watchmath said...

Since you also discuss matrices here, I think you should emphasize here that A and B are numbers. Because of course if A and B are matrices then AB=0 doesn't imply that A=0 or B=0.

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