Today we will look at the form मागधम् mAs from श्रीमद्भागवतम् 10.73.33.

गत्वा ते खाण्डवप्रस्थं शङ्खान्दध्मुर्जितारयः । हर्षयन्तः स्वसुहृदो दुर्हृदां चासुखावहाः ∥ १०-७३-३२ ∥
तच्छ्रुत्वा प्रीतमनस इन्द्रप्रस्थनिवासिनः । मेनिरे मागधं शान्तं राजा चाप्तमनोरथः ∥ १०-७३-३३ ∥

श्रीधर-स्वामि-टीका
खाण्डवप्रस्थमिन्द्रप्रस्थम् ∥ ३२ ∥ शान्तं मृतम् । आप्तमनोरथो बभूव ∥ ३३ ∥

Gita Press translation – On reaching the outskirts of the city, the three victorious heroes blew their respective conchs, bringing joy to their friends and sorrow to their enemies (32). The people of Indraprastha were extremely delighted at heart to hear the sound and concluded at once that Jarāsandha had been killed and that King Yudhiṣṭhira had well-nigh achieved his object (33).

The above verses have previously appeared in the following post – दुर्हृदाम्-mgp

मगधा जनपदः, मगधो नाम क्षत्रियः। मगधानां जनपदानां राजा = मागधः – a king of the territory Magadha, which is also the name of a kṣatriya. In the verses it refers to Jarāsandha.
Note: As a convention in the language, words (such as ‘मगध’) while denoting a territory are always used in the plural.

In the verses the विवक्षा is द्वितीया-एकवचनम्। Hence the form is मागधम्।

(1) मगध आम् + अण् । By 4-1-170 द्व्यञ्मगधकलिङ्गसूरमसादण् – To denote the sense of अपत्यम् (descendant) the तद्धित: affix ‘अण्’ may be applied optionally following a syntactically related पदम् which ends in a sixth case affix and has as its base either ‘मगध’ or ‘कलिङ्ग’ or ‘सूरमस’ or a word that has two vowels (अच् letters), provided the base denotes the name of a kṣatriya king, as well as the name of his country/territory.
Note: The affix ‘अण्’ prescribed by this सूत्रम् is अपवादः to the affix ‘अञ्’ prescribed by 4-1-168 जनपदशब्दात्‌ क्षत्रियादञ्।
As per the वार्तिकम् (under the सूत्रम् 4-1-168 जनपदशब्दात्‌ क्षत्रियादञ्) ’क्षत्रियसमानशब्दाज्जनपदात् तस्य राजन्यपत्यवत्’ – The same तद्धित: affix as is used to denote a descendant, may also be used to denote a king, when applied following a syntactically related पदम् which ends in a sixth case affix and has as its base a word that denotes the name of a country/territory, as well as the name of a kṣatriya.

Note: The सिद्धान्तकौमुदी says, तद्राजमाचक्षाणस्तद्राज इत्यन्वर्थसंज्ञासामर्थ्यात् – The above वार्तिकम् is inferred from the fact that the affixes ‘अञ्’ etc (prescribed by 4-1-168 etc) get the designation तद्राजाः (explained as तेषां राजानस्तद्राजाः) by the सूत्रम् 4-1-174 ते तद्राजाः, which implies that these affixes are also used to denote kings (of those countries/territories).
Note: The तत्वबोधिनी says, इह ‘4-2-125 अवृद्धादपि बहुवचनविषयात्’ इति प्राप्तो वुञ् बाध्यते – Here, the affix ‘अण्’ stops the affix ‘वुञ्’ which would have been prescribed by the सूत्रम् 4-2-125 अवृद्धादपि बहुवचनविषयात्।

(2) मगध आम् + अ । अनुबन्ध-लोपः by 1-3-3 हलन्त्यम्, 1-3-9 तस्य लोपः।
Note: ‘मगध आम् + अ’ gets the designation प्रातिपदिकम्‌ by 1-2-46 कृत्तद्धितसमासाश्च। This allows 2-4-71 सुपो धातुप्रातिपदिकयोः to apply in the next step.

(3) मगध + अ । By 2-4-71 सुपो धातुप्रातिपदिकयोः – A सुँप् affix takes a लुक् elision when it is a part of a धातुः or a प्रातिपदिकम्।

Note: The affix ‘अण्’ is a णित् (has the letter ‘ण्’ as इत्)। This allows 7-2-117 तद्धितेष्वचामादेः to apply in the next step.

(4) मागध + अ । By 7-2-117 तद्धितेष्वचामादेः – The first vowel among the vowels of a अङ्गम् which is followed by a तद्धित: affix (ref. 4-1-76 तद्धिताः) which is either a ञित् (has the letter ‘ञ्’ as a इत्) or a णित् (has the letter ‘ण्’ as a इत्) takes the वृद्धि: (ref. 1-1-1 वृद्धिरादैच्) substitute.
Note: The अङ्गम् ‘मागध’ gets the भ-सञ्ज्ञा by 1-4-18 यचि भम् which allows 6-4-148 यस्येति च to apply in the next step.

(5) मागध् + अ । By 6-4-148 यस्येति च – When the letter ‘ई’ or a ‘तद्धित’ affix follows, the (ending) इ-वर्ण: (letter ‘इ’ or ‘ई’) or the (ending) अवर्ण: (letter ‘अ’ or ‘आ’) of the अङ्गम् with the भ-सञ्ज्ञा takes elision (लोपः)।

= मागध । Note: The प्रातिपदिकम् ‘मागध’ declines like राम-शब्दः।

Similarly we can derive the following –
१) अङ्गानां जनपदानां राजा = आङ्गः।
२) कलिङ्गानां जनपदानां राजा = कालिङ्गः।
३) सूरमसानां जनपदानां राजा = सौरमसः।