Let A be a 3 3 matrix and (A)=2.

Question

Let A be a $3	imes 3$ matrix and $\det (A)=2$. If $n=\det (\underset{2024-}{\underset{}{adj(adj(\ldots ..(adjA))}}))$, then the remainder when $n$ is divided by 9 is equal to \_\_\_\_\_\_\_\_\_\_.

Answer: 7

Accepted Answer

$\begin{gathered} |A|=2 \ \underset{2024}{\underset{}{adj(adj(adj\ldots ..(a)))}}=|A{|}^{(n-1{)}^{2024}} \ =|A{|}^{2024} \ ={2}^{{2}^{2024}} \end{gathered}$

$\begin{gathered} {2}^{2024}=({2}^{2}){2}^{2022}=4(8{)}^{674}=4(9-1{)}^{674} \ \Rightarrow {2}^{2024}\equiv 4(mod9) \ \Rightarrow {2}^{2024}\equiv 9m+4,m\leftarrow \ {2}^{9m+4}\equiv 16\cdot {({2}^{3})}^{3m}\equiv 16(mod9) \ \, \equiv 7 \end{gathered}$

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