if a matrix M is given by `[{:(0,1,2),(1,2,3),(3,1,1):}]` and if `M[{:(alpha),(beta),(gamma):}]=[{:(1),(2),(3):}]` then

Options

A. `adj(M^(-1))+(adjM)^(-1)=-M`
B. `|adj(M^(2))|=81`
C. `alpha+2beta+3gamma=2`
D. `beta+2gamma=3`

Answer

A, C

Explanation

`[{:(0,1,2),(1,2,3),(3,1,1):}][{:(alpha),(beta),(gamma):}]=[{:(1),(2),(3):}]`
`impliesbeta+2gamma=1`
`alpha+2beta+3gamma=2`
`3alpha+beta+gamma=3`
`impliesalpha=1,beta=-1,gamma=1`
`|M|=-2`
`|adjM^(2)|=|M^(2)|^(2)=|M|^(4)=16`
`adj(M^(-1))=|M^(-1)|M=(-M)/(2)`
`(adjM)^(-1)=adj(M^-1)=-(M)/(2)`