Let `A` and `B` be symmetric matrices of same order. Then `A+B` is a symmetric matrix, `AB-BA` is a skew symmetric matrix and `AB+BA` is a symmetric matrix

Explanation

Given `A` and B are symmetric matrices

`Rightarrow-A^{T}=A` and `B^{T}=B`

(i) To prove `A B+B A` is a symmetric matrix.

Proof: Now `(A B+B A)^{T}=(A B)^{T}+(B A)^{T}=B^{T} A^{T}+A^{T} B^{T}`

`=B A+A B=A B+B A`

i.e. `(A B+B A)^{T}=A B+B A`


`Rightarrow(AB+BA)` is a symmetric matrix.

(ii) To prove `A B` - BA is a skew symmetric matrix.

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