If `A=[(-4,-1),(3,1)]` , then the determinant of the matrix `(A^2016 -2A^2015 -A^2014)` is :

Options

A. 2014
B. 2016
C. `-175`
D. `-25`

Answer

B

Explanation

Let `B=A^2016-2A^2015-A^2014`. Then,
`B=A^2014(A^2-2A-1)`
`rArr absB=absA^2014abs(A^2-2A-1)`
Now,`A=[(-4,-1),(3,1)]`
`:.{:A^2-2A-I=[(-4,-1),(3,1)][(-4,-1),(3,1)]-2[(-4,-1),(3,1)]-[(1,0),(0,1)]:}`
`{:= [(13,3),(-9,-2)]+[(8,2),(-6,-2)]+[(-1,0),(0,-1)]=[(20,5),(-15,-5)]:}`
`rArrabs(A^2-2A-I)={:[(20,5),(-15,-5)]=-25:}`
`and absA={:abs((-4,-1),(3,1))=-1:}`
Hence,`absB=(-1)^2014=-25`