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 $a_{23}$ is?

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

Answer 1

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

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

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

$|A|=-2-18$

$|A|=-20$

The cofactor matrix is $C=\left|\begin{array}{ccc}-2& -28& 18\\ -6& 7& -6\\ -4& -8& 4\end{array}\right|$

Thus, the adjoint of $A$ is $C^T$

Thus $A^{-1}=\frac{1}{-20}\left|\begin{array}{ccc}-2& -6& 4\\ -28& 7& -8\\ 18& -6& 4\end{array}\right|$

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