Let `A` be a square matrix. Then prove that `(i) A + A^T` is a symmetric matrix,`(ii) A -A^T` is a skew-symmetric matrix and`(iii) AA^T` and `A^TA` are symmetric matrices.

Explanation

(i) Let `P=A+A^(T)`.
`:. P^(T)=(A+A^(T))^(T)`
`=A^(T)+(A^(T))^(T)" "[ :' (A+B)^(T)=A^(T)+B^(T)]`
`=A^(T)+A" "[ :' (A^(T))^(T)=A]`
`=A+A^(T)" "[ :' "matrix addition is commutative"]`
`=P`
Therefore, P is a symmetric matrix.
(ii) Let `Q=A-A^(T)`.
`:. Q^(T)=(A-A^(T))^(T)`
`=A^(T)-(A^(T))^(T)`
`=A^(T)-A`
`=-(A-A^(T))`
`=-Q`
therefore, Q is skew-symmetric