Let A be a nxxn matrix and let B=A+ATandC=A−AT (a) Show that B is symmetric...

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.