If A is a square matric of order 5 and `2A^(-1)=A^(T)`, then the remainder when `|"adj. (adj. (adj. A))"|` is divided by 7 is

Options

A. 2
B. 3
C. 4
D. 5

Answer

A

Explanation

We have,
`2A^(-1)=A^(T)`
`implies |2A^(-1)|=|A^(T)|`
`implies 2^(5) 1/(|A|)=|A|`
`implies |A|^(2)=2^(5)`
`:. |"adj. (adj. (adj. A))"|=|A|^(4^(3))=(|A|^(2))^(32)=2^(160)`
`2^(160)/7=(2(2^(3))^(53))/7=(2(7+1)^(53))/7= m+2/7`, where m is integer
Thus, required remainder is 2.