Question

The inverse of the matrix $\left[\begin{array}{ccc}1& -2& -3\\ 0& 1& 2\\ 0& 0& 1\end{array}\right]$

• A. $\left[\begin{array}{ccc}1& 2& -1\\ 0& 1& -2\\ 0& 0& 1\end{array}\right]$
• B. $\left[\begin{array}{ccc}1& 2& 7\\ 0& 1& 2\\ 0& 0& 1\end{array}\right]$
• C. $\left[\begin{array}{ccc}1& 0& 0\\ 2& 1& 0\\ 7& 2& 1\end{array}\right]$
• D. $\frac{1}{2}\left[\begin{array}{ccc}1& 2& -7\\ 0& 1& 2\\ 0& 0& 1\end{array}\right]$

Answer 1

The correct option is A $\left[\begin{array}{ccc}1& 2& -1\\ 0& 1& -2\\ 0& 0& 1\end{array}\right]$

$A=\left[\begin{array}{ccc}1& -2& -3\\ 0& 1& 2\\ 0& 0& 1\end{array}\right]$
$|A|=1$
So $A^{-1}$ exists.
$A^{-1}=\frac{adjA}{|A|}$
Cofactor matrix $C=\left[\begin{array}{ccc}1& 0& 0\\ 2& 1& 0\\ -1& -2& 1\end{array}\right]$
$adjA=C^T=\left[\begin{array}{ccc}1& 2& -1\\ 0& 1& -2\\ 0& 0& 1\end{array}\right]$
$\Rightarrow A^{-1}=\left[\begin{array}{ccc}1& 2& -1\\ 0& 1& -2\\ 0& 0& 1\end{array}\right]$