Question

If $A=\left[\begin{array}{ccc}0& 1& 2\\ 1& 2& 3\\ 3& a& 1\end{array}\right]\text{and}{A}^{-1}=\left[\begin{array}{ccc}1/2& -1/2& 1/2\\ -4& 3& c\\ 5/2& -3/2& 1/2\end{array}\right],$ then $a+c=$

Answer 1

We have, $I=A{A}^{-1}=\frac{1}{2}\left[\begin{array}{ccc}0& 1& 2\\ 1& 2& 3\\ 3& a& 1\end{array}\right]\left[\begin{array}{ccc}1& -1& 1\\ -8& 6& 2c\\ 5& -3& 1\end{array}\right]$

$=\left[\begin{array}{ccc}1& 0& c+1\\ 0& 1& 2(c+1)\\ 4(1-a)& 3(a-1)& 2+ac\end{array}\right]$
$\Rightarrow a-1=0\Rightarrow a=1$
$\Rightarrow c+1=0\Rightarrow c=-1$
$\Rightarrow a+c=0$