Question

If $A^{-1}=\left[\begin{array}{ccc}1& -1& 2\\ 0& 3& 1\\ 0& 0& -1/3\end{array}\right],$ then

• A. $|A|=-1$
• B. $\text{adj}A=\left[\begin{array}{ccc}-1& 1& -2\\ 0& -3& -1\\ 0& 0& 1/3\end{array}\right]$
• C. $A=\left[\begin{array}{ccc}1& 1/3& 7\\ 0& 1/3& 1\\ 0& 0& -3\end{array}\right]$
• D. $A=\left[\begin{array}{ccc}1& -1/3& -7\\ 0& -3& 0\\ 0& 0& 1\end{array}\right]$

Answer 1

Answer: (A), (B), (C)

$A=\left[\begin{array}{ccc}1& 1/3& 7\\ 0& 1/3& 1\\ 0& 0& -3\end{array}\right]$

$|A^{-1}|=-1\Rightarrow |A|=-1$
Now, $\text{adj}A=|A|A^{-1}$
$\text{adj}A=\left[\begin{array}{ccc}-1& 1& -2\\ 0& -3& -1\\ 0& 0& 1/3\end{array}\right]$
Also, we know $A=(A^{-1}{)}^{-1}$
$\Rightarrow A=\frac{\left(\text{adj}{A}^{-1}\right)}{|A^{-1}|}=\left[\begin{array}{ccc}1& 1/3& 7\\ 0& 1/3& 1\\ 0& 0& -3\end{array}\right]$