Question # Let A be ann xx n matrix and let B = A + A^T and C = A − A^T (a) Show that B is symmetric and C is skew symmetric. (b) Show that every n × n matrix can be represented as a sum of a symmetric matrix and a skew-symmetric matrix.

Matrices
ANSWERED Let A be a $$n xx n$$ matrix and let
$$B = A + A^T and C = A − A^T$$
(a) Show that B is symmetric and C is skew symmetric.
(b) Show that every n × n matrix can be represented as a sum of a symmetric matrix and a skew-symmetric matrix. 2021-02-09
as we know that a matrix E is said to be symmetric if $$E^T=E$$ and skew symmetric if $$E^T=−E.$$
as $$B=A+A^T$$ therefore,
$$=B^T =(A+A^T)^T$$
$$=A^T+(A^T)^T$$
$$=A^T+A (As (A^T)^T=A )$$
=B
as $$B^T=B$$ therefore, B is symmetric matrix.
hence proved.
As C=A−AT
therefore, $$CT=(A−A^T)^T$$
$$=A^T−(A^T)T$$
$$=A^T−A$$
$$=−(A−A^T)$$
=−C
as $$C^T=−C$$ therefore, C is skew symmetric matrix. Hence proved.
Now we have to show that every nxxn matrix can be expressed as the sum of the symmetric and skew symmetric matrix. Let A be nxxn matrix. therefore,
$$A=1/2(2A)$$
$$=1/2(A+A)$$
$$=1/2(A+A^T+A−A^T)$$
$$=1/2(B+C)$$
where $$B=A+A^T$$ and B is a symmetric matrix and $$C=A−A^T$$ and C is a skew symmetric matrix.
herefore,
$$A=1/2(B+C)$$
$$=1/2$$(symmetric matrix +skew symmetric matrix)
therefore, it has been showed that any matrix A of order nxxn can be expressed as the sum of symmetric and skew symmetric matrix.