Which of the following statements is/are correct If A and B are two symmetric matrices of order n then 1. A + B is also a symmetric matrix. 2. AB is a symmetric matrix

Which of the following statements is/are correct If A and B are two symmetric matrices of order n then 1. A + B is also a symmetric matrix. 2. AB is a symmetric matrix Correct Answer Only 1

Concept:

• Symmetric Matrix: Any real square matrix A = (a_ij) is said to be symmetric matrix if and only if a_ij = a_ji, ∀ i and j or in other words we can say that if A is a real square matrix such that A = A’ then A is said to be a symmetric matrix.
• (A ± B)' = A' ± B'
• (A ⋅ B)' = B' ⋅ A'

Calculation:

Given: A and B are two symmetric matrices of order n

Statement 1: A + B is also a symmetric matrix.

Let's find out transpose of A + B

⇒ (A + B)' = A' + B'

∵ A and B are two symmetric matrices of order n i.e A' = A and B' = B

⇒ (A + B)' = A + B

Hence, statement 1 is true.

Statement 2: A ⋅ B is a symmetric matrix

Let's find out the transpose of A ⋅ B

⇒ (A ⋅ B)' = B' ⋅ A'

∵ A and B are two symmetric matrices of order n i.e A' = A and B' = B

⇒ (A ⋅ B)' = B ⋅ A

But we also know that matrix multiplication is not commutative in general

So, we cannot say that (A ⋅ B)' = A ⋅ B in general

Hence, statement 2 is false.