A and B are square matrices such that det. `(A)=1, B B^(T)=I`, det `(B) gt 0`, and `A( adj. A + adj. B)=B`.
The value of det `(A+B)` is

Options

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

Answer

D

Explanation

`B B^(T)=I`
`:. |B B^(T)|=1`
`implies |B||B^(T)|=1`
`implies |B||B^(T)|=1`
`implies |B|=1`
A(adj. A + adj. B)=B
`implies |A|I+Axx"adj "B=B`
`implies I+Axx"adj "B=B`
`implies IxxB+Axx("adj B")xxB=B^(2)`
`implies B+A|B|I=B^(2)`
`implies B+A=B^(2)` (1)
`implies |A+B|=|B|^(2)=1`