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. -2
• B. -1
• C. 2
• D. 5

Answer 1

Answer: (D)

5

$A=\left[\begin{array}{ccc}1& -1& 1\\ 2& 1& -3\\ 1& 1& 1\end{array}\right]$

Matrix of cofactor of $A==\left[\begin{array}{ccc}4& -5& 1\\ 2& 0& -2\\ 2& 5& 3\end{array}\right]$

Adjoint of this matrix $=\left[\begin{array}{ccc}4& 2& 2\\ -5& 0& 5\\ 1& -2& 3\end{array}\right]$

And $\left|A\right|=\left|\begin{array}{ccc}1& -1& 1\\ 2& 1& -3\\ 1& 1& 1\end{array}\right|$

$=1(1+3)+1(2+3)+1(2-1)=10$

$\therefore$ Inverse of $A=\frac{1}{10}\left[\begin{array}{ccc}4& 2& 2\\ -5& 0& 5\\ 1& -2& 3\end{array}\right]$

Now as inverse of $A$ is equal to $B$

$\therefore \frac{1}{10}\left[\begin{array}{ccc}4& 2& 2\\ -5& 0& 5\\ 1& -2& 3\end{array}\right]=\frac{1}{10}\left[\begin{array}{ccc}4& 2& 2\\ -5& 0& \alpha \\ 1& -2& 3\end{array}\right]$

$\Rightarrow \alpha =5$