If A and B are symmetric matrices and AB = BA, then A<sup>-1</sup>B is a -
Step-by-Step Solution
Key Concept: A matrix M is symmetric if M^T = M. Given A^T = A, B^T = B, and AB = BA. We need to check if (A^-1B)^T = A^-1B.
Given A and B are symmetric, so A<sup>T</sup> = A and B<sup>T</sup> = B. Also AB = BA. Let M = A<sup>-1</sup>B. Then M<sup>T</sup> = (A<sup>-1</sup>B)<sup>T</sup> = B<sup>T</sup>(A<sup>-1</sup>)<sup>T</sup> = B(A<sup>T</sup>)<sup>-1</sup> = BA<sup>-1</sup>. Since AB = BA, multiplying by A<sup>-1</sup> on both sides gives A<sup>-1</sup>AB A<sup>-1</sup> = A<sup>-1</sup>BAA<sup>-1</sup>, which simplifies to BA<sup>-1</sup> = A<sup>-1</sup>B. Thus M<sup>T</sup> = M, so A<sup>-1</sup>B is symmetric.
Correct Answer: 1