Find `(AB)^(-1) " if "A = {:((1,2,3),(1,-2,-3)):}, B = {:((1,-1),(1,2),(1,-2)):}`

Explanation

We have, `H=[{:(1,2,3),(1,-2,-3):}]and B=[{:(1,-1),(1,2),(1,-2):}]`
Now `AB=[{:(1,2,3),(1,-2,-3):}][{:(1,-1),(1,2),(1,-2):}]`
`AB=[{:(1+2+3,-1+4-6),(1-2-3,-1-4+6):}]`
`AB=[{:(6-3),(-41):}`
Now, we get `M_(11)=1,`
`A_(11)=(-1)^(1+1)(1)=1`
`M_(12)=-4,`
`A_(12)=(-1)^(1+2)(-4)=4`
`M_(21)=(-1)^(1+2)(-3)=3`
`M_(22)=6,`
`A_(22)=(-1)^(2+2)(6)=6`
therefeforeadj.`(AB)=[{:(1,4),(3,6):}]^(T)=[{:(1,3),(4,6):}]`
and `|AB|=|{:(1,3),(4,6):}|=6-12=-6ne0`
`therefore` Using `A^(-1)=(1)/(|A|)adj. A`
`therefore(AB)^(-1)=-1/6[{:(1,3),(4,6):}]` which is the required inverse.