Let $A =[a_{ij}]$ be a 3 × 3 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 cofactors of elements of matrix $A_{n-1}$, then $|A_n|$ equals

$|A|^{2^n}$

Clearly,

$|A_1|=|A|^2,|A_2|=|A_1|^2, |A_3|=|A_2|^2,..., |A_n|=|A_{n-1}|^2$

$∴|A_n|=(|A_{n-2}|^2)^2$  $[∵|A_{n-1}|=|A_{n-2}|^2]$

$⇒|A_n|=(|A_{n-2}|)^{2×2}$

$⇒|A_n|=(|A_{n-3}|^2)^{2×2}=|A_{n-3}|^{2×2×2}$

$⇒|A_n|=....=|A|^{2×2×2×....×2n-times}=|A|^{2^n}$