If `A=[(0,c,-b),(-c,0,a),( b,-a,0)]` and `B=[(a^2,a b, a c),( a b,b^2,b c ),(a c ,b c,c^2)]` , show that `A B=B A=O_3xx3` .

Explanation

given,
`A=[(0,c,-b),(-c,0,a),( b,-a,0)]`

`B=[(a^2,a b, a c),( a b,b^2,b c ),(a c ,b c,c^2)]`
`AB=[(0,c,-b),(-c,0,a),( b,-a,0)]xx[(a^2,a b, a c),( a b,b^2,b c ),(a c ,b c,c^2)]`
`=[(0+abc−abc,0+b^2c-b^2c,0+bc^2-bc^2),(-ca^2+0+ca^2,-abc+0+abc,-ac^2+0+ac^2),(ba^2-a^2b+0,ab^2-ab^2+0,abc-abc+0)]`
`=[(0,0,0),(0,0,0),(0,0,0)]`
`BA=[(a^2,a b, a c),( a b,b^2,b c ),(a c ,b c,c^2)]xx[(0,c,-b),(-c,0,a),( b,-a,0)]`
`=[(0+abc−abc,0+a^2c-a^2c,0+ba^2-ba^2),(-cb^2+0+cb^2,-abc+0+abc,-ab^2+0+ab^2),(bc^2-c^2b+0,ac^2-ac^2+0,abc-abc+0)]`
`=[(0,0,0),(0,0,0),(0,0,0)]`
`AB=BA=0_(3xx3)`