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

• A. $a=1,c=-1$
• B. $a=2,c=\frac{1}{2}$
• C. $a=1,c=2$
• D. $a=\frac{1}{2},c=\frac{3}{2}$

Answer 1

Answer: (A)

$a=1,c=-1$

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

$\left(A{A}^{-1}\right)=I$

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

$=0\left(\frac{1}{2}\right)+1\left(c\right)+2\left(\frac{1}{2}\right)$

$0+c+1=0\Rightarrow c=-1$

$\frac{3}{2}-4\left(a\right)+\frac{5}{2}=0\Rightarrow 4a=4\Rightarrow a=1$

$a=1,c=-1$