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Is it necessary that there should be matter at the centre of mass of system?


No, it is not necessary that there be matter at the centre of mass of the system.

For e.g., if two equal point masses are separated by certain distance, the centre of mass lies at the mid point of two point masses and there is no mass at that point.
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What is the significance of defining the center of mass of a system?

The motion of n particle system can be reduced to one particle motion.

An equivalent single point object would enable us to discuss the overall motion of the system. 
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A system consists of two particles of masses mand mhaving position vectors 
rightwards arrow for straight F subscript 1 of space and space rightwards arrow for straight r subscript 2. of. What is the position vector of the centre of mass of system?


Given, the system consists of two masses m1 and m2. Position vectors of the masses m1 and m2 are r1 and r2 respectively.

Therefore, the position vector of the centre of mass of the system is given by, 

                  

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Under what conditions the centre of mass of two bodies lies midway between the two bodies?

When the masses of two bodies are equal then centre of mass lies midway between the two bodies.
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A body at rest explodes in three fragments. What is the velocity of the centre of mass of exploded part?

Zero.

In the absence of external force, the centre of mass moves with constant velocity. Since before explosion the body was at rest, therefore, the velocity of the centre of mass is zero before an explosion. Hence it will also be zero after the explosion.
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