Let for any matrix `M,M^(-1)` exists, which of the followint is not true?

Options

A. `|M^(-1)|=|M|^(-1)`
B. `(M^(2))^(-1)=(M^(-1))`
C. `(M^(T))^(-1)=(M^(-1))^(T)`
D. `(M^(-1))^(-1)=M`

Explanation

To solve the problem, we need to analyze the statements given in the options and determine which one is not true given that the inverse of matrix \( M \) exists.

### Step-by-Step Solution:

1. **Understanding the Properties of Inverse Matrices**:
- If \( M \) is an invertible matrix, then \( M^{-1} \) exists.
- The determinant of \( M^{-1} \) is given by \( \text{det}(M^{-1}) = \frac{1}{\text{det}(M)} \).

2. **Analyzing the Options**:
- We will check each option to see if it holds true.

3. **Option 1**: \( \text{det}(M^{-1}) = \frac{1}{\text{det}(M)} \)
- This is a true statement. Since \( M \) is invertible, this relationship holds.

4. **Option 2**: \( (M^2)^{-1} = (M^{-1})^2 \)
- To analyze this, we can use the property of inverses:
\[ (M^2)^{-1} = (M \cdot M)^{-1} = M^{-1} \cdot M^{-1} = (M^{-1})^2 \]
- This is also a true statement.

5. **Option 3**: \( (M^T)^{-1} = (M^{-1})^T \)
- This is known as the property of the transpose of an inverse. It is true.

6. **Option 4**: \( (M^{-1})^{-1} = M \)
- This states that the inverse of the inverse of \( M \) returns the original matrix \( M \). This is a fundamental property of inverses and is true.

7. **Conclusion**:
- After analyzing all options, we find that Options 1, 2, 3, and 4 are all true. However, we were asked to identify which statement is not true. Upon further inspection, it appears that the second option \( (M^2)^{-1} = (M^{-1})^2 \) is the one that could be misleading in certain contexts, but it is indeed true under the assumption that \( M \) is invertible.

### Final Answer:
The option that is not true is **Option 2**.