Okay, let's tackle this question about the first dorsal interosseous muscle of the hand. The question is asking which nerve supplies this muscle. First, I need to recall the anatomy of the hand muscles and their innervation.

The dorsal interosseous muscles are part of the intrinsic muscles of the hand. There are four of them, and they're located between the metacarpal bones. Each dorsal interosseous muscle has a specific function and nerve supply. The first one, between the first and second metacarpals, is the one in question here.

I remember that the dorsal interosseous muscles are innervated by the deep branch of the ulnar nerve. Wait, but let me confirm. The ulnar nerve splits into superficial and deep branches. The deep branch (also called the dorsal interosseous nerve) specifically supplies these muscles. The first dorsal interosseous muscle is between the thumb and index finger metacarpals, so its supply should be from the deep ulnar nerve.

The other options might include the radial nerve, median nerve, or maybe the superficial branch of the ulnar nerve. The radial nerve supplies the extensor muscles of the forearm, not the intrinsic hand muscles. The median nerve innervates the thenar muscles and some lumbricals, but not the dorsal interossei. The superficial branch of the ulnar nerve is sensory, providing sensation to the medial hand, not motor function.

So the correct answer should be the deep branch of the ulnar nerve. Let me make sure I didn't mix up with the palmar interosseous muscles. The palmar interossei are also supplied by the ulnar nerve, but the question specifically mentions the dorsal ones. Wait, no, actually, all interossei (both dorsal and palmar) are supplied by the deep ulnar nerve. So the first dorsal interosseous muscle is indeed innervated by the deep ulnar nerve. Therefore, the correct answer is the deep ulnar nerve.

**Core Concept**
The first dorsal interosseous muscle is an intrinsic hand muscle innervated by the **deep branch of the ulnar nerve**. This nerve arises from the medial and lateral cords of the brachial plexus (C8-T1) and provides motor supply to most intrinsic hand muscles, excluding those innervated by the median or radial nerves.

**Why the Correct Answer is Right**
The deep branch of the ulnar nerve (dorsal interosseous nerve) supplies the first dorsal interosseous muscle. This muscle originates from the first and second metacarpals and inserts into the base of the first proximal phalanx. It functions to adduct the index finger. The ulnar nerve’s deep branch travels between the flexor and extensor compartments of the forearm before entering the hand, where it divides into palmar and dorsal terminal branches. The dorsal branch specifically innervates the dorsal interossei and other extensor muscles.

**Why Each Wrong Option is Incorrect**
**Option A:** *Median nerve* – Incorrect. The median nerve supplies thenar muscles (e.g., flexor pollicis brevis) but not dorsal interossei.
**Option B:** *Radial nerve* – Incorrect. The radial nerve innervates extensor muscles of the forearm and dorsal hand, not intrinsic hand

✓ Correct Answer: A. Deep branch of ulnar nerve