Question

If $A^{-1}=\left[\begin{array}{ccc}1& 0& -2\\ -2& 1& 0\\ -1& 1& 0\end{array}\right],$ then which of the following options are correct?

• A. $\left|A\right|=2$
• B. $adj\left(A\right)=\left[\begin{array}{ccc}\frac{1}{2}& 0& -1\\ -1& \frac{1}{2}& 0\\ -\frac{1}{2}& \frac{1}{2}& 0\end{array}\right]$
• C. $\left|adj\left(A\right)\right|=4$
• D. $\left|A^{\prime }\right|=\frac{1}{2}$

Answer 1

Answer: (B), (D)

The correct options are
B $\left|A^{\prime }\right|=\frac{1}{2}$
D $adj\left(A\right)=\left[\begin{array}{ccc}\frac{1}{2}& 0& -1\\ -1& \frac{1}{2}& 0\\ -\frac{1}{2}& \frac{1}{2}& 0\end{array}\right]$

We are given $A^{-1}=\left|\begin{array}{ccc}1& 0& -2\\ -2& 1& 0\\ -1& 1& 0\end{array}\right|$

$\Rightarrow \left|A^{-1}\right|=-2(-2+1)=2$

We check all the options

A. $\left|A\right|=\frac{1}{\left|A^{-1}\right|}=\frac{1}{2}$

B. $adj\left(A\right)=A^{-1}\left|A\right|=\frac{1}{2}\left|\begin{array}{ccc}1& 0& -2\\ -2& 1& 0\\ -1& 1& 0\end{array}\right|=\left|\begin{array}{ccc}\frac{1}{2}& 0& -1\\ -1& \frac{1}{2}& 0\\ -\frac{1}{2}& \frac{1}{2}& 0\end{array}\right|$

C. $\left|adj\left(A\right)\right|=-1(-\frac{1}{2}+\frac{1}{4})=\frac{1}{4}$

D. $\left|A^{{}^{\prime }}\right|=\left|A\right|=\frac{1}{2}$

B,D are correct