Question

For two non-singular matrices A&B, show that adj (AB)=adj(AB)=(adjB)(adjA)

Answer 1

We have (AB)(adj AB)=`|AB|I_(n)`
`|A|=|B| I_(n)`
`A^(-1)(AB)("adj"AB))=|A||B|A^(-1)`
`Rightarrow B"adj"(AB)=|B|adjA (therefore A^(-1)(1)/(|A|)adjA)`
`Rightarrow B^(-1)B"adj"(AB)=|B|B^(-1)adjA`
`Rightarrow B^(-1)B "adj"(AB)|B|B^(-1)"adj"A`
`Rightarrow "adj"(AB)=("adj"B)("adjA")`