Let A and B be real invertible matrices such that AB = −BA. Then

Question 1Answer

a.
Trace(A) =Trace(B) = 1


b.
Trace(A) = 1, Trace(B) = 0


c.
Trace(A) =Trace(B) = 0


d.
Trace(A) = 0, Trace(B) = 1

Given that \( A \) and \( B \) are real invertible matrices such that \( AB = -BA \), we want to find the correct relationship between their traces.

### Analyzing the Property \( AB = -BA \)

1. **Trace of a Matrix:**
   The trace of a matrix \( X \), denoted as \( \text{Tr}(X) \), is the sum of its diagonal elements. A useful property of the trace function is that for any two matrices \( X \) and \( Y \), 
   \[
   \text{Tr}(XY) = \text{Tr}(YX).
   \]

2. **Applying the Trace to the Given Equation:**
   Given \( AB = -BA \), we can take the trace of both sides:

   \[
   \text{Tr}(AB) = \text{Tr}(-BA).
   \]

   Using the property of the trace, we have:

   \[
   \text{Tr}(AB) = \text{Tr}(-BA) = -\text{Tr}(BA).
   \]

   Since \( \text{Tr}(AB) = \text{Tr}(BA) \), this gives:

   \[
   \text{Tr}(AB) = -\text{Tr}(AB).
   \]

   This implies:

   \[
   \text{Tr}(AB) = 0.
   \]

3. **Conclusion on Traces of \( A \) and \( B \):**
   Since \( AB = -BA \) and both \( A \) and \( B \) are invertible, \( A \) and \( B \) must be matrices with specific properties. In particular, if they are real matrices and satisfy the anti-commutation relation \( AB = -BA \), their traces must satisfy:

   \[
   \text{Tr}(A) = 0 \quad \text{and} \quad \text{Tr}(B) = 0.
   \]

   This is because the trace of a product of such matrices will always yield zero. Thus, both matrices have to be traceless.

### Correct Answer:

The correct answer is:

**c. \( \text{Tr}(A) = \text{Tr}(B) = 0 \).**

Would you like more details or have any questions? Here are 5 related questions that could further enhance understanding:

1. Why is the trace of a product of matrices invariant under cyclic permutations?
2. Can you provide an example of two \(2 \times 2\) matrices \( A \) and \( B \) such that \( AB = -BA \) and both have a trace of zero?
3. What are the implications of a matrix being traceless in terms of its eigenvalues?
4. How does the property \( AB = -BA \) influence the determinant of the matrices \( A \) and \( B \)?
5. Are there any geometric interpretations for matrices that satisfy \( AB = -BA \)?

**Tip:** When dealing with matrix equations, always consider fundamental properties like trace, determinant, and rank for insight.

Let A and B be real invertible matrices such that AB = −BA. Then

a. Trace(A) =Trace(B) = 1

b. Trace(A) = 1, Trace(B) = 0

c. Trace(A) =Trace(B) = 0

d. Trace(A) = 0, Trace(B) = 1

Explore the solution to the problem involving real invertible matrices A and B, where AB = −BA. Learn how the traces of A and B must satisfy Tr(A) = 0 and Tr(B) = 0 through the application of matrix theory and trace properties. Suitable for college-level linear algebra students.

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