Question

Let A be a $3\times 3$ matrix such that
$adjA=\left[\begin{array}{ccc}2& -1& 1\\ -1& 0& 2\\ 1& -2& -1\end{array}\right]$ and
B = adj (adj A)
If $|A|=\lambda$ and $\left|{\left(B^{-1}\right)}^T\right|=\mu$, then the ordered pair, $(|\lambda |,\mu )$ is equal to :

• A. $\left(9,\frac{1}{9}\right)$
• B. $\left(9,\frac{1}{81}\right)$
• C. $\left(3,\frac{1}{81}\right)$
• D. $(3,81)$

Answer 1

Answer: (C)

$C=adjA=\left[\begin{array}{ccc}+2& -1& 1\\ -1& 0& 2\\ 1& -2& -1\end{array}\right]$
$|C|=|adjA|=+2(0+4)+1.(1-2)+1.(2,4)$
$=+8-1+2$
$|adjA|=|A{|}^2=9=9$
$\lambda =|A|=\pm 3$
$|\lambda |=3$
$B=adjC$
$|B|=|adjC|=|C{|}^2=81$
$\left|{\left(B^{-1}\right)}^T\right|=|B{|}^{-1}=\frac{1}{81}$
$(|\lambda |,\mu )=\left(3,\frac{1}{81}\right)$