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

Answer: (B)

1.-1

Given

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

We know,

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

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

$\Rightarrow \left[\begin{array}{ccc}-4+5& 3-3& c+1\\ 1/2-8+\frac{15}{2}& 1/2+6-\frac{9}{2}& 1/2+2c+\frac{3}{2}\\ 3/2+(-4a)+5/2& 3/2+3a-\frac{3}{2}& 3/2+ac+1/2\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

$\Rightarrow \left[\begin{array}{ccc}1& 0& \alpha +1\\ 0& 6-\frac{8}{2}& 2c+2\\ 4-4a& 3a& ac+2\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

Comparing each entry of the matrices on both sides, we get

$c+1=0$

$\Rightarrow c=-1$

and $4-4a=0$

$\Rightarrow 4a=4$

$\Rightarrow a=1$

$\therefore a=1,c=-1$.