Question

If $A=\left[\begin{array}{ccc}3& -3& 4\\ 2& -3& 4\\ 0& -1& 1\end{array}\right]$ then $A^{-1}$ equals to =?

• A. $A^2$
• B. $A^4$
• C. $A$
• D. $A^3$

Answer 1

The correct option is D $A^3$

$A=\left[\begin{array}{ccc}3& -3& 4\\ 2& -3& 4\\ 0& -1& 1\end{array}\right]$
Here, $|A|=3\left(1\right)+3\left(2\right)+4(-2)=1$
Now, $adjA=C^T={\left[\begin{array}{ccc}1& -2& -2\\ -1& 3& 3\\ 0& -4& -3\end{array}\right]}^T$
$\Rightarrow adjA=\left[\begin{array}{ccc}1& -1& 0\\ -2& 3& -4\\ -2& 3& -3\end{array}\right]$
$\Rightarrow A^{-1}=\left[\begin{array}{ccc}1& -1& 0\\ -2& 3& -4\\ -2& 3& -3\end{array}\right]$
Now, $A^2=\left[\begin{array}{ccc}3& -3& 4\\ 2& -3& 4\\ 0& -1& 1\end{array}\right]\left[\begin{array}{ccc}3& -3& 4\\ 2& -3& 4\\ 0& -1& 1\end{array}\right]$
$\Rightarrow A^2=\left[\begin{array}{ccc}3& -4& 4\\ 0& -1& 0\\ -2& 2& -3\end{array}\right]$
Now, $A^3=A^{2}A=\left[\begin{array}{ccc}3& -4& 4\\ 0& -1& 0\\ -2& 2& -3\end{array}\right]\left[\begin{array}{ccc}3& -3& 4\\ 2& -3& 4\\ 0& -1& 1\end{array}\right]$
$\Rightarrow A^3=\left[\begin{array}{ccc}1& -1& 0\\ -2& 3& -4\\ -2& 3& -3\end{array}\right]$
Hence, $A^{-1}=A^3$