Question

Let $A=\left[\begin{array}{ccc}1& 2& -1\\ -1& 1& 2\\ 2& -1& 1\end{array}\right]$ and B is non-singular matrix. Then which one of the following is INCORRECT

• A. $|adj\left(adjA\right)|={14}^4$
• B. $|B^{-1}AB|=14$
• C. $|2A|=112$
• D. $A.adj\left(A\right)=14\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

Answer 1

The correct option is A $|adj\left(adjA\right)|={14}^4$

$\begin{array}{c}A=\left(\begin{array}{ccc}1& 2& -1\\ -1& 1& 2\\ 2& -1& 1\end{array}\right)\\ \Rightarrow \left|A\right|=1\left(3\right)-2\left(-5\right)-\left(-1\right)\\ \Rightarrow \left|A\right|=14\\ \left|adj\left(adjA\right)\right|={\left|A\right|}^{{\left(h-1\right)}^2}={14}^{2^2}={14}^4\\ \left|B^{-1}AB\right|=14\\ \left|B^{-1}BA\right|=14\\ \left|IA\right|=14\left[\begin{array}{c}R{R}^{-1}=I\\ IA=I\end{array}\right]\\ \left|A\right|=14\\ \Rightarrow \left|2A\right|=2^3\left|A\right|=8\times 14=112\end{array}$