If B is the inverse of matrix A, then a is
A) 5
B) -1
C) 2
D) -2
Correct Answer: A
Solution :
\[A=\left| \begin{matrix} 1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1 \\ \end{matrix} \right|\] 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 \[C=\left| \begin{matrix} 4 & -5 & 1 \\ 2 & 0 & -2 \\ 2 & 5 & 3 \\ \end{matrix} \right|\] \[\therefore \]Adj \[A={{C}^{T}}=\left| \begin{matrix} 4 & 2 & 2 \\ -5 & 0 & 5 \\ 1 & -2 & 3 \\ \end{matrix} \right|\] \[\therefore \]\[{{A}^{-1}}=\frac{AdjA}{|A|}=\frac{1}{10}\left| \begin{matrix} 4 & 2 & 2 \\ -5 & 0 & 5 \\ 1 & -2 & 3 \\ \end{matrix} \right|\] Comparing, we get \[\alpha =5\]
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