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If the matrix AB is a zero matrix, then which one of the following is correct?

  1. A must be equal to zero matrix or B must be equal to zero matrix
  2. A must be equal to zero matrix and B must be equal to zero matrix
  3. It is not necessary that either A is zero matrix or B is zero matrix
  4. None of the above

Answer (Detailed Solution Below)

Option 3 : It is not necessary that either A is zero matrix or B is zero matrix
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Detailed Solution

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Concept:

  and  But AB = 0.

Calculations:

Given,  the matrix AB is a zero matrix

⇒ AB = 0

Consider,  and 

⇒ and 

Now 

⇒AB = 0

Here,  and  but AB = 0.

Hence, if the matrix AB is a zero matrix, then It is not necessary that either A is zero matrix or B is zero matrix.

 

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