If square matrices A and B are such that `"AA"^(theta),"BB"^(theta)=B^(theta)B,"AB"^(theta)=B^(theta)A`, then `"AB"("AB")^(theta)` is equal to

Options

A. `"ABA"^(theta)B^(theta)`
B. `A^(theta)B^(theta)BA`
C. `(AB)^(theta)AB`
D. none of these

Answer

To solve the problem, we need to find the expression \( AB(AB)^\theta \) given the conditions on matrices \( A \) and \( B \). Let's break it down step by step.

### Step 1: Understand the Given Conditions
We have the following conditions:
1. \( AA^\theta = B^\theta B \)
2. \( AB^\theta = B^\theta A \)

### Step 2: Rewrite the Expression
We need to find \( AB(AB)^\theta \). We can express this as:
\[ AB(AB)^\theta = AB(B^\theta A^\theta) \]
This uses the property that \( (XY)^\theta = Y^\theta X^\theta \) for matrices \( X \) and \( Y \).

### Step 3: Substitute Using the Given Conditions
From the second condition \( AB^\theta = B^\theta A \), we can substitute \( AB^\theta \) in our expression:
\[ AB(AB)^\theta = AB(B^\theta A^\theta) = A(BB^\theta)A^\theta \]

### Step 4: Use the First Condition
From the first condition \( AA^\theta = B^\theta B \), we can substitute \( BB^\theta \):
\[ AB(AB)^\theta = A(AA^\theta)A^\theta \]
Now, substituting \( AA^\theta \):
\[ AB(AB)^\theta = A(B^\theta B)A^\theta \]

### Step 5: Simplify the Expression
Now, we can simplify the expression:
\[ AB(AB)^\theta = AB^\theta A \]
Using the second condition again, we know \( AB^\theta = B^\theta A \):
\[ AB(AB)^\theta = B^\theta A A^\theta \]

### Step 6: Final Result
Since \( A A^\theta \) is a symmetric matrix, we can conclude:
\[ AB(AB)^\theta = B^\theta (AA^\theta) \]
Using the first condition again, we know \( AA^\theta = B^\theta B \):
\[ AB(AB)^\theta = B^\theta (B^\theta B) = B^\theta B^\theta B \]

Thus, the final result is:
\[ AB(AB)^\theta = B^\theta B \]