Let A=[[|gathered 2 1 0 1 2 -1 0 -1 2 gathered]]|.

Question

Let $A=[[|\begin{gathered} 2 \quad  1 \quad  0 \ 1 \quad  2 \quad  -1 \ 0 \quad  -1 \quad  2 \end{gathered}]]|$. If $|adj(adj(adj2A))|=(16{)}^{n}$, then $n$ is equal to :

Accepted Answer

We have,

$\begin{gathered} |A|=|\begin{array}{ccc}2 \quad  1 \quad  0 \ 1 \quad  2 \quad  -1 \ 0 \quad  -1 \quad  2\end{array}|=2(4-1)-1(2-0)+0 \ =6-2=4 \ |2A|={2}^{3}|A|=8	imes 4=32 \ |adj(adj(adj2A))|=|2A{|}^{(n-1{)}^{3}} \ =(32{)}^{{2}^{3}}={32}^{8} \ \Rightarrow {16}^{n}=(32{)}^{8}={2}^{8}	imes {16}^{8} \ \Rightarrow {16}^{n}={16}^{2+8}\Rightarrow n=10 \end{gathered}$

Concepts :

(a) $|kA|={k}^{n}|A|$

(b) $|adjA|=|A{|}^{n-1}$

Solutions · 1