# Let A be a 3 x 3 matrix such that

Question:

Let $A$ be a $3 \times 3$ matrix such that

adj $A=\left[\begin{array}{ccc}2 & -1 & 1 \\ -1 & 0 & 2 \\ 1 & -2 & -1\end{array}\right]$ and

$B=\operatorname{adj}(\operatorname{adj} A)$

If $|\mathrm{A}|=\lambda$ and $\left|\left(\mathrm{B}^{-1}\right)^{\mathrm{T}}\right|=\mu$, then the ordered pair, $(|\lambda|, \mu)$ is equal to :

1. $\left(9, \frac{1}{9}\right)$

2. $\left(9, \frac{1}{81}\right)$

3. $\left(3, \frac{1}{81}\right)$

4. $(3,81)$

Correct Option: , 3

Solution:

$\mathrm{C}=\operatorname{adj} \mathrm{A}=\left|\begin{array}{ccc}+2 & -1 & 1 \\ -1 & 0 & 2 \\ 1 & -2 & -1\end{array}\right|$

$|C|=|\operatorname{adj} A|=+2(0+4)+1 .(1-2)+1 .(2,4)$

$=+8-1+2$

$|\operatorname{adj} \mathrm{A}|=|\mathrm{A}|^{2}=9=9$b

$\lambda=|\mathrm{A}|=\pm 3$

$|\lambda|=3$

$\mathrm{B}=\operatorname{adj} \mathrm{C}$

$|B|=|\operatorname{adj} C|=|C|^{2}=81$

$\left|\left(\mathrm{B}^{-1}\right)^{\mathrm{T}}\right|=|\mathrm{B}|^{-1}=\frac{1}{81}$

$(|\lambda|, \mu)=\left(3, \frac{1}{81}\right)$