Suppose A and B are two `3xx3` matrices with entries from complex numbers such that ABA = I. Which one the following is not true ?

Options

A. B is invertible
B. `B^(-1) =A^(2)`
C. A is not invertible
D. `A^(4)B^(2)=I`

Answer

To solve the problem, we need to analyze the given equation \( ABA = I \) where \( A \) and \( B \) are \( 3 \times 3 \) matrices, and \( I \) is the identity matrix. We will derive properties of \( A \) and \( B \) and determine which statement is not true based on these properties.

### Step 1: Understand the given equation
We start with the equation:
\[ ABA = I \]
This implies that \( A \) and \( B \) are related in such a way that the product of \( A \), \( B \), and \( A \) results in the identity matrix.

### Step 2: Take the determinant of both sides
Taking the determinant on both sides gives us:
\[ \text{det}(ABA) = \text{det}(I) \]
Since the determinant of the identity matrix \( I \) is 1, we have:
\[ \text{det}(A) \cdot \text{det}(B) \cdot \text{det}(A) = 1 \]
This simplifies to:
\[ (\text{det}(A))^2 \cdot \text{det}(B) = 1 \]

### Step 3: Analyze the implications
From the equation \( (\text{det}(A))^2 \cdot \text{det}(B) = 1 \), we can conclude:
1. \( \text{det}(A) \neq 0 \) (since if \( \text{det}(A) = 0 \), the left-hand side would be 0, contradicting the right-hand side which is 1).
2. \( \text{det}(B) \neq 0 \) (since if \( \text{det}(B) = 0 \), the left-hand side would again be 0).

Thus, both \( A \) and \( B \) are invertible matrices.

### Step 4: Find the inverse of \( B \)
To find the inverse of \( B \), we can manipulate the original equation \( ABA = I \):
1. Post-multiply both sides by \( A^{-1} \):
\[ ABA A^{-1} = I A^{-1} \implies AB = A^{-1} \]
2. Now, pre-multiply both sides by \( A^{-1} \):
\[ A^{-1} AB = A^{-1} A^{-1} \implies B = A^{-1} A^{-1} = (A^{-1})^2 \]

### Step 5: Analyze the statements
Now we need to analyze the statements provided in the options to determine which one is not true. Based on our findings:
- \( A \) and \( B \) are both invertible.
- The inverse of \( B \) is \( A^2 \).

### Conclusion
The statement that is not true among the options provided would be the one that contradicts the invertibility of \( A \) or \( B \) or incorrectly states the relationship between \( A \) and \( B \).