Question

Let A and B be symmetric matrices of same order, then which of the following statement is true?

• A. (A+B) is a symmetric matrix
• B. (AB) is a symmetric matrix
• C. (A+B) is a skew-symmetric matrix
• D. (AB) is a skew-symmetric matrix

Answer 1

The Correct Option is A

To determine which statement about matrices A and B is true, when A and B are symmetric matrices of the same order, we start by exploring the properties of symmetric matrices.

Definition of Symmetric Matrix: A matrix \( M \) is symmetric if \( M = M^T \), where \( M^T \) is the transpose of \( M \).

Given:

A and B are symmetric matrices, hence:

A = A^T and B = B^T.

Let's consider the options:

• (A+B) is a symmetric matrix: For this to be true, \((A+B)^T\) must equal \((A+B)\).

Compute the transpose of the sum:

\((A+B)^T = A^T + B^T\)

Since A and B are symmetric, this simplifies to:

\(A^T + B^T = A + B\)

Hence, \( (A+B) \) is indeed symmetric.

• (AB) is a symmetric matrix: For AB to be symmetric, \((AB)^T\) must equal \(AB\).

Let's compute the transpose:

\((AB)^T = B^T A^T\)

Since A and B are symmetric, this becomes:

\(B A\)

In general, \(BA \neq AB\), so \( (AB) \) is not symmetric unless A and B commute, which is not given.

• (A+B) is a skew-symmetric matrix: A matrix \( M \) is skew-symmetric if \( M^T = -M \).

\((A+B)\) is shown to be symmetric, so it cannot be skew-symmetric.

• (AB) is a skew-symmetric matrix: This would require \((AB)^T = -(AB)\).

As shown above, \((AB)^T = BA\), and does not generally equal \(-(AB)\).

Hence, (AB) is not skew-symmetric.

Conclusively, option (A+B) is a symmetric matrix is true.