HOW IS THE SELF IMMUTABLE AND  IMMORTAL?

na  jaayate mriyate vaa kadaachin
    naayam  bhootwaa bhavitaa vaa na bhooyah
    ajo  nityah shaashwato'yam puraano
    na  hanyate hanyamaane shareere //2.20//

He  is never born nor does He ever die; after having been, He again does not cease  to be.  Unborn, eternal, changeless and  ancient. He is not killed when the body is killed.

This  verse describes the absence of the six kinds of modification inherent in every  living thing viz., birth, subsistence, growth, transformation, decay and death.  The Self is altogether changeless. These changes are the source of all sorrows  and miseries in every mortal's life.  All  these are denied to the Self to prove Its changelessness.  Birth and death are for the physical bodies  only and they cannot touch the immortal Self just as the waves are born and die  in the ocean but the ocean itself is not born with the waves nor does it die  when the waves disappear.

Arjuna  was grieved about the death of his kinsmen in the war. So the Lord explains  that the soul is not killed when the body is slain and hence he should not  grieve.

WHY THE SOUL DOES NOT KILL  ANYBODY?

vedaavinaashinam  nityam ya enam ajam avyayam
    katham  sa purushah paartha kam ghaatayati hanti kam   // 2.21 //

Whosoever  knows Him to be indestructible, eternal, unborn and immutable, how can that man  slay O Arjuna, or cause another to slay?

An  enlightened person who knows the changelessness and the indestructibility of  the Self cannot perform the function of slaying or cause another to slay. When  we know the Self to be invulnerable, how can anyone slay it?  The words ‘how can he slay' refer to Arjuna  and `cause another to slay’ refer to Krishna's own role.

Summarizing  what has been said so far Krishna emphasizes that those who know the nature of  the Self shall have no dejection or sorrow in the face of the realities of  life. Therefore, one while discharging duty should not grieve, while slaying  anyone or causing anyone to be slain, but should discharge one’s duty, in  accordance with the ordinance of scriptures.

ILLUSTRATION REGARDING  INDESTRUCTIBILITY OF THE SOUL.

vaasaamsi  jeernaani yathaa vihaaya
    navaani  grihnaati naro'paraani
    tathaa  shareeraani vihaaya jeernaa
    nyanyaani  samyaati navaani dehee // 2.22 //
    Just  as a man casts off his worn-out clothes and puts on new ones, so also the  embodied Self casts off Its worn out bodies and enters that are new.

Here 'dehi' means the jiva - the individual person, who is made up of  the perceptible gross physical body, the imperceptible subtle and causal  bodies, together with the atman.
  The  verse says:  Just as an individual person  gives up worn out or old clothing and takes up new  ones, similarly, the same jiva, on giving up  the worn out or old gross physical body,  naturally takes up an appropriate new gross  physical body. By giving up the old clothes and putting on new ones, the person  does not change. Similarly, by giving up the old body and assuming a new one,  the atman in the jiva - the individual person - does not change.   The `worn out condition of the body' does not  refer to its biological condition but to the capacity of the body, mind and  intellect equipments to earn the required experiences from the available  environment for facilitating their evolutionary journey. This evolution and  change is for the physical bodies and not for the Self.

The  verse also tells something more about every individual person. By virtue of one's  own 'karma' the jiva already becomes ready to assume a new body, prior to  casting out the old worn-out body which has served its purpose.  In other words, the mental make-up of a person  does not die along with the death of the gross body. The mental make-up of the  person, along with its karma-born tendencies and dispositions is called the  subtle-body which is the core of every jiva and it survives the death of the  physical frame. In its next step of evolution, the jiva assumes a new physical  frame more suited to the fulfillment of its natural tendencies and  dispositions. In all these changes, the self or soul or the Atman remains  unchanged. In reality, the soul being immobile and non-active does not migrate  from one body to another; it is ever fixed and steady and does not undergo any  change whatsoever.

But  just as when a pot is carried from one place to another, the space within the  pot also appears be carried, even so when the subtle body leaves a gross body  and enters another, it appears that the soul also has moved from one body to  another. Therefore, the acts of leaving one body and entering into another are  attributed to the soul in order to explain the phenomenon of death to the lay  people. The word ‘dehi’ is indicative of the soul identifying itself  with gross body it appears to be leaving and entering into another. In this  sense it is said that the soul leaves a worn-out body and enters into a new  one.

Now  a question arises why this cycle of birth and death has been going on from  times immemorial. While this question can be answered from the Jnana, Bhakti  and Karma points of view, the basic factor behind this never-ending cycle is  that God has granted the choice to the living beings to make proper use of  their lives and to rediscover ultimately their own transcendental nature.  Through innumerable births in the relative world they gain experience, through  experience knowledge and through knowledge attain freedom or liberation or moksha from this cycle.