If matrix `A=[(0,1,-1),(4,-3,4),(3,-3,4)]=B+C`, where B is symmetric matrix and C is skew-symmetric matrix, then find matrices B and C.

Answer

`B=1/2 [(0,5,2),(5,-6,1),(2,1,8)], C=1/2[(0,-3,-4),(3,0,7),(4,-7,0)]`

Explanation

Here matrix A is expressed as the sum of symmetric and skew-symmetric matrix. Then
`B=1/2(A+A^(T))` and `C=1/2 (A-A^(T))`
Now `A=[(0,1,-1),(4,-3,4),(3,-3,4)]`
`implies A^(T)=[(0,4,3),(1,-3,-3),(-1,4,4)]`
`implies B=1/2 ([(0,1,-1),(4,-3,4),(3,-3,4)]+[(0,4,3),(1,-3,-3),(-1,4,4)])`
`=1/2 [(0,5,2),(5,-6,1),(2,1,8)]`
and `C=1/2 ([(0,1,-1),(4,-3,4),(3,-3,4)]-[(0,4,3),(1,-3,-3),(-1,4,4)])`
`=1/2 [(0,-3,-4),(3,0,7),(4,-7,0)]`