Let A be an inbertible matrix. Which of the following is not true?

Options

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

Explanation

To determine which statement is not true regarding the invertible matrix \( A \), we will analyze each option step by step.

### Step-by-Step Solution:

1. **Understanding Invertible Matrices**:
An invertible matrix \( A \) is one that has an inverse denoted by \( A^{-1} \) such that:
\[ A \cdot A^{-1} = I \]
where \( I \) is the identity matrix.

2. **Option A**: \( (A^T)^{-1} = (A^{-1})^T \)
- This is a known property of matrices. The inverse of the transpose of a matrix is equal to the transpose of the inverse of that matrix. Thus, this statement is **true**.

3. **Option B**: \( A^{-1} = \det(A^{-1}) \)
- Here, \( A^{-1} \) is a matrix, while \( \det(A^{-1}) \) is a scalar (a number). A matrix cannot equal a scalar. Therefore, this statement is **not true**.

4. **Option C**: \( A^{-2} = (A^2)^{-1} \)
- To check this, we can multiply both sides by \( A^2 \):
\[ A^{-2} \cdot A^2 = I \quad \text{and} \quad (A^2)^{-1} \cdot A^2 = I \]
- Both sides yield the identity matrix, confirming that this statement is **true**.

5. **Option D**: \( \det(A^{-1}) = \frac{1}{\det(A)} \)
- This is another known property of determinants. The determinant of the inverse of a matrix is equal to the reciprocal of the determinant of the matrix. Thus, this statement is **true**.

### Conclusion:
The only statement that is not true is **Option B**: \( A^{-1} = \det(A^{-1}) \).

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