If `A=[(1,0,0),(0,1,1),(0,-2,4)] , 6A^(-1)=A^2+cA+dI`, then (c,d) is :

Options

A. (-6,11)
B. (-11,6)
C. (11,6)
D. (6,11)

Answer

A

Explanation

Every square matrix A satisfies its characteristic equation i.e. `abs(A-lambdal)=0`.
Here, `abs(A_lambdal)=0`
`{:rArr abs((1-lambda,0,0),(0,1-lambda,1),(0,-2,4-lambda))=0:}`
`rArr (1-lambda){(1-lambda)(4-lambda)+2}=0`
`rArr lambda^3-6lambda^2+11lambda-6=0`
`rArr A^3 -6A^2+11A-6l=0`
`rArr 6I=A^3-6A^2+11A`
`rArr 6A^(-1)=A^2-6A+11I` [Multiplying both sides by `A^(-1)`]
`:. 6A^(-1) =A^2=cA+dl rArrc=-6and d=11`