To solve the problem related to the matrix \( M \), we need to follow the steps outlined in the question:

### a. Find the conditions for the matrix \( M \) to be defined:

The matrix \( M \) includes several rational functions. For \( M \) to be defined, we need to identify the values of \( x \) that do not cause any denominators to equal zero.

1. **First term:** \(\frac{x^2}{x^3 - 4x}\)
Denominator: \(x^3 - 4x = x(x^2 - 4) = x(x - 2)(x + 2)\)
This term is undefined for \( x = 0, 2, -2 \).

2. **Second term:** \(\frac{6}{6 - 3x}\)
Denominator: \(6 - 3x = 0 \Rightarrow x = 2\).
This term is undefined for \( x = 2 \).

3. **Third term:** \(\frac{1}{x + 2}\)
Denominator: \(x + 2 = 0 \Rightarrow x = -2\).
This term is undefined for \( x = -2 \).

4. **Last term in the multiplication:** \(\frac{10 - x^2}{x + 2}\)
Same condition as before, undefined for \( x = -2 \).

### Summary of restrictions:
- **The function \( M \) is undefined for:** \( x = 0, 2, -2 \).

### b. Simplify the matrix \( M \):

We can simplify \( M \) by substituting the various expressions and performing algebraic operations. However, we need the complete rational expression to provide a clear simplification. Let's denote \( N(x) = x - 2 + \frac{10 - x^2}{x + 2} \) which can be simplified separately.

### c. Calculate the value of \( M \) when \( |x| = \frac{1}{2} \):

We will evaluate \( M \) at \( x = \frac{1}{2} \) and observe if it is within the defined conditions. If it is valid, we substitute \( x = \frac{1}{2} \) into the simplified expression of \( M \) to find its value.

### Summary:

1. Conditions for \( M \) to be defined: \( x \neq 0, 2, -2 \).
2. To simplify \( M \), substitute and combine terms.
3. Value at \( |x| = \frac{1}{2} \): substitute \( x = \frac{1}{2} \) in the simplified \( M \).

If you have further instructions on how to simplify or compute \( M \), please provide the explicit form of \( M \).