Question

If `A=[{:(3,2),(1,0):}],B=[{:(1,-2,5),(0,7,3):}]" and "C=[{:(8,1,-6),(2,-5,0):}]`, verify that `A(B+C)=(AB+AC).`

Answer 1

We have
`A(B+C)=[{:(3,2),(1,0):}].{[{:(1,-2,5),(0," "7,3):}]+[{:(8,1,-6),(2,-5," "0):}]}`
`=[{:(3,2),(1,0):}][{:(9,-1,-1),(2," "2," "3):}]`
`=[{:(3.9+2.2,3.(-1)+2.2,3.(-1)+2.3),(1.9+0.2,1.(-1)+.2,1.(-1)+0.3):}]`
`=[{:(31," "1," "3),(9,-1,-1):}].`
Now, `AB=[{:(3,2),(1,0):}][{:(1,-2,5),(0,7,3):}]`
`=[{:(3.1+2.0,3.(-2)+2.7,3.5+2.3),(1.1+0.0,1.(-2)+0.7,1.5+0.3):}]`
`=[{:(3,8,21),(1,-2,5):}].`
And, `AC=[{:(3,2),(1,0):}][{:(8,1,-6),(2,-5," "0):}]`
`=[{:(3.8+2.2,3.1+2.(-5),3.(-6)+2.0),(1.8+0.2,1.1+0.(-5),1.(-6)+0.0):}]`
`=[{:(28,-7,-18),(8,1,-6):}].`
`:." "(AB+AC)=[{:(3,8,21),(1,-2,5):}]+[{:(28,-7,-18),(8,1,-6):}]`
`=[{:(31," "1," "3),(9,-1,-1):}].`
Hence, `A(B+C)=AB+AC.`