Question

Let $A=\left[\begin{array}{ccc}1& -1& 1\\ 2& 1& -3\\ 1& 1& 1\end{array}\right]$ and $(10)B=\left[\begin{array}{ccc}4& 2& 2\\ -5& 0& \alpha \\ 1& -2& 3\end{array}\right]$ , If $B$ is the inverse of matrix $A$ , then $\alpha$ is

• A. $5$
• B. $-1$
• C. $2$
• D. $-2$

Answer 1

Answer: (A)

$A=\left[\begin{array}{ccc}1& -1& 1\\ 2& 1& -3\\ 1& 1& 1\end{array}\right]$
Cofactors of various entries are
$4,-5,1;2,0,-2;2,5,3$
$|A|=1\times 4+(-1)\times -5+1\times 1=10$
Cofactor Matrix $C=\left[\begin{array}{ccc}4& -5& 1\\ 2& 0& -2\\ 2& 5& 3\end{array}\right]$
$\therefore AdjA=C^T=\left[\begin{array}{ccc}4& 2& 2\\ -5& 0& 5\\ 1& -2& 3\end{array}\right]$
$\therefore A^{-1}=\frac{\text{ Adj}\text{A}}{|A|}=\frac{1}{10}\left[\begin{array}{ccc}4& 2& 2\\ -5& 0& 5\\ 1& -2& 3\end{array}\right]$
Comparing, we get $\alpha =5$