Question

If A and B are two non-singular matrices of order 3 such that `A A^(T)=2I` and `A^(-1)=A^(T)-A`. Adj. `(2B^(-1))`, then det. (B) is equal to
A. 4
B. `4sqrt(2)`
C. 16
D. `16sqrt(2)`

Answer 1

Correct Answer - D
`A A^(T)=2I` ...(1)
`implies |A|^(2)=8`
Now, `A^(-1)=A^(T)-A" adj."(2B^(-1))` ...(2)
Multiplying with A, we get
`I=A A^(T)-A^(2)"adj."(2B^(-1))`
`:. I=2I-A^(2)"adj."(2B^(-1))`
`implies A^(2)"adj." (2B^(-1))=I`
`implies |A^(2)||"adj." (2B^(-1))|=1`
`implies 8|2B^(-1)|^(2)=1`
`implies 8. 64/(|B|^(2))=1`
`implies |B|^(2)=64xx8`
`:. |B|= pm (8xx2sqrt(2))= pm 16 sqrt(2)`