Question

Let A be a 3 x 3 matrix such that
$A\left[\begin{array}{ccc}1& 2& 3\\ 0& 2& 3\\ 0& 1& 1\end{array}\right]=\left[\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right]$
Then $A^{-1}$ is :

• A. $\left[\begin{array}{ccc}3& 1& 2\\ 3& 0& 2\\ 1& 0& 1\end{array}\right]$
• B. $\left[\begin{array}{ccc}3& 2& 1\\ 3& 2& 0\\ 1& 1& 0\end{array}\right]$
• C. $\left[\begin{array}{ccc}0& 1& 3\\ 0& 2& 3\\ 1& 1& 1\end{array}\right]$
• D. $\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 1\\ 0& 2& 3\end{array}\right]$

Answer 1

Answer: (A)

given $A\left[\begin{array}{ccc}1& 2& 3\\ 0& 2& 3\\ 0& 1& 1\end{array}\right]=\left[\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right]$
use column transformation and make RHS as I
$\left(i\right)C_1\leftrightarrow C_{3}A\left[\begin{array}{ccc}3& 2& 1\\ 3& 2& 0\\ 1& 1& 0\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]$
$\left(ii\right)C_2\leftrightarrow C_{3}A\left[\begin{array}{ccc}3& 1& 2\\ 3& 0& 2\\ 1& 0& 1\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
$A^{-1}=\left[\begin{array}{ccc}3& 1& 2\\ 3& 0& 2\\ 1& 0& 1\end{array}\right]$