Question

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

• A. $-1,1$
• B. $1,2$
• C. $1,-1$
• D. None

Answer 1

The correct option is C $1,-1$

$A=\left[\begin{array}{ccc}0& 1& 2\\ 1& 2& 3\\ 3& a& 1\end{array}\right]$ $A^{-1}=\left[\begin{array}{ccc}1/2& -1/2& 1/2\\ -4& 3& C\\ 5/2& -3/2& 1/2\end{array}\right]$

$A{A}^{-1}=I$

$\left[\begin{array}{ccc}0& 1& 2\\ 1& 2& 3\\ 3& a& 1\end{array}\right]$ $\left[\begin{array}{ccc}1/2& -1/2& 1/2\\ -4& 3& c\\ 5/2& -3/2& 1/2\end{array}\right]$ $=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

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

$c+1=0$

$c=-1$

$4-4a=0$

$a=1$