Question

If $A=|a_{ij}{|}_{2\times 2}$, where $a_{ij}=\{\begin{array}{ccc}i+j,& if& i\ne j\\ i^2-2j,& if& i=j\end{array}$, then $A^{-1}=?$

• A. $\frac{1}{9}\left[\begin{array}{cc}0& 3\\ 3& 1\end{array}\right]$
• B. $\frac{1}{9}\left[\begin{array}{cc}0& -3\\ -3& -1\end{array}\right]$
• C. $\frac{1}{9}\left[\begin{array}{cc}4& 1\\ -1& 2\end{array}\right]$
• D. $\frac{-1}{9}\left[\begin{array}{cc}4& 1\\ -1& 2\end{array}\right]$

Answer 1

The correct option is A $\frac{1}{9}\left[\begin{array}{cc}0& 3\\ 3& 1\end{array}\right]$

$\begin{array}{c}A={\left|aij\right|}_{2\times 2}\\ a_{11}=1-2=-1\\ a_{12}=3\\ a_{21}=3\\ a_{22}=4-2\times 2=0\\ A=\left[\begin{array}{cc}-1& 3\\ 3& D\end{array}\right]\\ A^{-1}=\frac{1}{\left|A\right|}adjA\\ =\frac{-1}{09}\left[\begin{array}{cc}0& -3\\ -3& -1\end{array}\right]\\ =\frac{1}{9}\left[\begin{array}{cc}0& 3\\ 3& 1\end{array}\right]\end{array}$