Question

Let A and B be real symmetric matrices of same size. Which one of the following options is correct?

• A. \( (AB)^T = B^T A^T \)
• B. \( AB = BA \)
• C. \( A^T = A^{-1} \)
• D. \( A = A^{-1} \)

Hint:

Remember the fundamental properties of matrix operations. The reversal rule for transpose, \( (AB)^T = B^T A^T \), and for inverse, \( (AB)^{-1} = B^{-1} A^{-1} \), are always true for any conforming matrices.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question provides that A and B are real symmetric matrices and asks to identify the correct statement among the given options. A matrix M is symmetric if \(M^T = M\).
Step 2: Detailed Explanation:
Let's analyze each option:

• (A) \( (AB)^T = B^T A^T \): This is the general "reversal rule" for the transpose of a product of matrices. This property is true for any two matrices A and B for which the product AB is defined. It does not depend on whether the matrices are symmetric or not. Since it is a universally true mathematical identity, it is correct in this specific case as well.
• (B) \( AB = BA \): This states that the matrices commute. The product of two symmetric matrices is symmetric if and only if they commute. However, symmetric matrices do not commute in general. For example, let \( A = \begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix} \) and \( B = \begin{pmatrix} 3 & 4 \\ 4 & 5 \end{pmatrix} \). Both are symmetric.
  \( AB = \begin{pmatrix} 11 & 14 \\ 10 & 13 \end{pmatrix} \) and \( BA = \begin{pmatrix} 11 & 10 \\ 14 & 13 \end{pmatrix} \). Clearly, \( AB \neq BA \). So, this option is incorrect.
• (C) \( A^T = A^{-1} \): This is the definition of an orthogonal matrix. A symmetric matrix is not necessarily orthogonal. For example, the matrix \( A = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} \) is symmetric, but \( A^{-1} = \begin{pmatrix} 1/2 & 0 \\ 0 & 1/2 \end{pmatrix} \), which is not equal to \( A^T = A \). So, this is incorrect.
• (D) \( A = A^{-1} \): This is the definition of an involutory matrix, where \( A^2 = I \). A symmetric matrix is not necessarily involutory. Using the same example \( A = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} \), we have \( A \neq A^{-1} \). So, this is incorrect.

Step 3: Final Answer:
The only statement that is always true is the general property of matrix transposition, \( (AB)^T = B^T A^T \). The other statements are conditions that apply only to specific subsets of symmetric matrices.