Question

Let $A^{-1}=\left[\begin{array}{ccc}1& 2017& 2\\ 1& 2017& 4\\ 1& 2018& 8\end{array}\right]$. Then , $|2A|-|2A^{-1}|$ is equal to

• A. 3
• B. -3
• C. 12
• D. -12

Answer 1

Answer: (C)

Given,
$A^{-1}=\left[\begin{array}{ccc}1& 2017& 2\\ 1& 2017& 4\\ 1& 2018& 8\end{array}\right]$
$\left|A^{-1}\right|-\left[\begin{array}{ccc}1& 2017& 2\\ 1& 2017& 4\\ 1& 2018& 8\end{array}\right]$
Apply $R_1\to R_1-R_2$
$\left|A^{-1}\right|=\left[\begin{array}{ccc}0& 0& -2\\ 1& 2017& 4\\ 1& 2018& 8\end{array}\right]$
Expand along $R_1$
$\left|A^{-1}\right|=-2\left(2018-2017\right)=-2$
$A{A}^{-1}=1$
$\left|A\right|\left|A^{-1}\right|=1$
$\left|A\right|=\frac{1}{{|A^{-1}|}^2}=-\frac{1}{2}$
Now, $\left|2A\right|-\left|2A^{-1}\right|=8\left|A\right|-8\left|A^{-1}\right|$
$=8\left[\frac{-1}{2}+2\right]=12$