Question

Let A be a $3 \times 3$ matrix such that
$adj A= \begin{bmatrix}2&-1&1\\ -1&0&2\\ 1&-2&-1\end{bmatrix}$ 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)

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