`A=[(3,a,-1),(2,5,c),(b,8,2)]` is symmetric and `B=[(d,3,a),(b-a,e,-2b-c),(-2,6,-f)]` is skew-symmetric, then find AB.

Answer

`AB=[(-4,3,-6),(-31,54,-26),(-28,9,-50)]`

Explanation

A is symmetric
`implies A^(T)=A`
`implies [(3,2,b),(a,5,8),(-1,c,2)]=[(3,a,-1),(2,5,c),(b,8,2)]`
`implies a=2, b=-1, c=8`
B is skew-symmetric
`implies B^(T)=-B`
`implies [(d,b-a,-2),(3,e,6),(a,-2b-c,-f)]=[(-d,-3,-a),(a-b,-e,2b+c),(2,-6,f)]`
`implies d=-d, f=-f` and `e=-e`
`implies d=f=0`
So `A=[(3,2,-1),(2,5,8),(-1,8,2)]` and `B=[(0,3,2),(-3,0,-6),(-2,6,0)]`
`implies AB=[(3,2,-1),(2,5,8),(-1,8,2)][(0,3,2),(-3,0,-6),(-2,6,0)]`
`=[(-4,3,-6),(-31,54,-26),(-28,9,-50)]`