Question

If matrix $A=\left[\begin{array}{ccc}1& 0& -1\\ 3& 4& 5\\ 0& 6& 7\end{array}\right]$ and its inverse is denoted by $A^{-1}=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$, then the value of $a_{23}=$

• A. $\frac{21}{20}$
• B. $\frac{1}{5}$
• C. $-\frac{2}{5}$
• D. $\frac{2}{5}$

Answer 1

The correct option is D $\frac{2}{5}$

$A=\left[\begin{array}{ccc}1& 0& -1\\ 3& 4& 5\\ 0& 6& 7\end{array}\right]$

$A^{-1}=\frac{adjA}{detA}$

$detA=1(28-30)-1\left(18\right)$

$=-2-18$

$=-20$

$adjA=\left[\begin{array}{ccc}-2& -6& +4\\ -21& 7& -8\\ 18& -6& 4\end{array}\right]$

$A^{-1}=\frac{-1}{20}\left[\begin{array}{ccc}-2& -6& 4\\ -21& 7& -8\\ 18& -6& 4\end{array}\right]$

$a_{23}=\frac{-8}{-20}=\frac{2}{5}$

$\therefore a_{23}=2/5$.