Express the matrix B=`[{:(,2,-2,),(,-1,3,):}]` as the sum of a symmetric and a skew symmetric matrix.

Explanation

Here, `B=[{:(,2,-1,1),(,-2,3,-2),(,-4,4,-3):}]`
`"Let P"=(1)/(2)(B+B')=(1)/(2)[{:(,4,-3,-3),(,-3,6,2),(,-3,2,-6):}]=[{:(,2,-(3)/(2),-(3)/(2)),(,-(3)/(2),3,1),(,-(3)/(2),1,-3):}]`
`"Now P"=[{:(,2,-(3)/(2),-(3)/(2)),(,-(3)/(2),3,1),(,-(3)/(2),1,-3):}]=P`
Thus, `P=(1)/(2)(B+B')` is a symmetric matrix.
Also, let `"Q"=(1)/(2)(B-B')=(1)/(2)[{:(,0,-1,-5),(,1,0,6),(,5,-6,0):}]=[{:(,0,-(1)/(2),-(5)/(2)),(,-(1)/(2),0,3),(,(5)/(2),-3,0):}]`
`"Now"Q'=[{:(,0,(1)/(2),(5)/(2)),(,-(1)/(2),0,-3),(,-(5)/(2),3,0):}]=-Q`
Thus, `Q=(1)/(2)(B-B')` is a skew symmetric matrix.
Now, `"Now"P+Q'=[{:(,2,(-3)/(2),(-3)/(2)),(,(-3)/(2),3,1),(,(-3)/(2),1,-3):}]+[{:(,0,(-1)/(2),(-5)/(2)),(,(1)/(2),0,3),(,(5)/(2),-3,0):}]=[{:(,2,-2,-4),(,-1,3,4),(,1,-2,4),(,1,-2,-3):}]`
Thus, B is represeted as the sum of a symmetric and a skew symmetric matrix.