Question

If $A=\left[\begin{array}{ccc}0& 1& 2\\ 1& 2& 3\\ 3& a& 1\end{array}\right]$ 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 the values of $a$ and $c$ are equal to

• A. $1,1$
• B. $1,-1$
• C. $1,2$
• D. $-1,1$

Answer 1

The correct option is B $1,-1$

$A=\left[\begin{array}{ccc}0& 1& 2\\ 1& 2& 3\\ 3& a& 1\end{array}\right]$ and $A^{-1}=\left[\begin{array}{ccc}1/2& 1/2& 1/2\\ -4& 3& c\\ 5/2& -3/2& 1/2\end{array}\right]$
Here, $|A|=-1(-8)+2(a-6)=2a-4$
$adjA=C^T={\left[\begin{array}{ccc}2-3a& 8& a-6\\ 2a-1& -6& 3\\ -1& 2& -1\end{array}\right]}^T$
$\Rightarrow adjA=\left[\begin{array}{ccc}2-3a& 2a-1& -1\\ 8& -6& 2\\ a-6& 3& -1\end{array}\right]$
Hence, $A^{-1}=\frac{1}{2a-4}\left[\begin{array}{ccc}2-3a& 2a-1& -1\\ 8& -6& 2\\ a-6& 3& -1\end{array}\right]$
Comparing with given $A^{-1}$,
$-\frac{1}{2a-4}=\frac{1}{2}$
$\Rightarrow a=1$
Also, $\frac{2}{2a-4}=c$
$\Rightarrow c=-1$
Thus the values of $a$ and $c$ are $1$ and $-1$ respectively.