`A` is a `2xx2` matrix such that `A[1-1]=[-1 2]a n dA^2[1-1]=[1 0]dot` The sum of the elements of `A` is `-1` b. 0 c. 2 d. 5

Options

A. `-1`
B. 0
C. 2
D. 5

Answer

D

Explanation

`A[(1),(-1)]=[(-1),(2)]` (1)
`A^(2)[(1),(-1)]=[(1),(0)]` (2)
Let a be given by `A=[(a,b),(c,d)]`. The first equation gives
`a-b=-1` (3)
`c-d=2` (4)
For the second equation,
`A^(2)[(1),(-1)]=A(A[(1),(-1)])=A([(-1),(2)])=[(1),(0)]`
This gives
`-a+2b=1` (5)
`-c+2d=0` (6)
Eqs, (3) + (5) gives `b=0` and `a=-1`
Eqs. (4) + (6) gives `d=2` and `c=4`
So the sum `a+b+c+d=5`.