Let `A = [a_(ij)] " be a " 3 xx3` matrix and let `A_(1)` denote the matrix of the cofactors of elements of matrix A and `A_(2)` be the matrix of cofactors of elements of matrix `A_(1)` and so on. If `A_(n)` denote the matrix of cofactros of elements of matrix `A_(n -1)`, then `|A_(n)|` equals

Options

A. `|A|^(2n)`
B. `|A|^(2n)`
C. `|A|^(n^(2))`
D. `|A|^(2)`

Answer

B

Explanation

Clearly,
`|A_(1)| = |A|^(2), |A_(2)| = |A_(1)|^(2),..., |A_(n)| = |A_(n-1)|^(2)`
`:. |A_(n)| = (|A_(n-2)|^(2))^(2) " " [ :' |A_(n-1)| = |A_(n-2)|^(2)]`
`rArr |A_(n)| = (|A_(n -2)|)^(2xx2)`
`rArr |A_(n)| = (|A_(n-3)|^(2))^(2xx2) = |A_(n-3)|^(2 xx 2 xx2)`
`rArr |A_(n)| = ...= |A|^(2 xx 2 xx 2 xx...xx 2n- " times") = |A|^(2n)`