Let A and B are `3xx3` matrices with real number entries, where A is symmetric, B is skew - symmetric and `(A+B)(A-B)=(A-B)(A+B)`. If `(AB)^(T)=(-1)^(k)AB`, then the sum of all possible integral value of k in `[2, 10]` is equal to (where `A^(T)` represent transpose of matrix A)

Options

A. 20
B. 24
C. 28
D. 45

Answer

To solve the problem, we need to analyze the given conditions step by step.

### Step 1: Understand the properties of matrices A and B
- Matrix A is symmetric, which means \( A^T = A \).
- Matrix B is skew-symmetric, which means \( B^T = -B \).

### Step 2: Analyze the equation \((A + B)(A - B) = (A - B)(A + B)\)
We can expand both sides of the equation.

**Left-hand side:**
\[ (A + B)(A - B) = A^2 - AB + BA - B^2 \]

**Right-hand side:**
\[ (A - B)(A + B) = A^2 + AB - BA - B^2 \]

### Step 3: Set the two expansions equal to each other
Equating the left-hand side and right-hand side:
\[ A^2 - AB + BA - B^2 = A^2 + AB - BA - B^2 \]

### Step 4: Simplify the equation
Cancel \( A^2 \) and \( -B^2 \) from both sides:
\[ -AB + BA = AB - BA \]
Rearranging gives:
\[ 2BA = 2AB \]
Dividing both sides by 2:
\[ BA = AB \]

### Step 5: Analyze the condition \((AB)^T = (-1)^k AB\)
Using the properties of transposes:
\[ (AB)^T = B^T A^T \]
Substituting the properties of A and B:
\[ (AB)^T = (-B)A = -BA \]
From Step 4, we know \( BA = AB \), so:
\[ (AB)^T = -AB \]

### Step 6: Set the equations equal
We have:
\[ -AB = (-1)^k AB \]
This implies:
\[ (-1)^k = -1 \]
This holds true when \( k \) is odd.

### Step 7: Find the integral values of k in the range [2, 10]
The odd integers in the range [2, 10] are:
- 3
- 5
- 7
- 9

### Step 8: Calculate the sum of these values
Now, we sum these odd integers:
\[ 3 + 5 + 7 + 9 = 24 \]

### Final Answer
The sum of all possible integral values of \( k \) in the range [2, 10] is **24**.