Question

Find the adjoint of ${\left(\frac{7^{-4}}{4^{-2}}\right)}^{\frac{1}{4}}\left[\begin{array}{ccc}1& -1& 2\\ 3& 0& -2\\ 1& 0& 3\end{array}\right]$

Answer 1

Let

$A={\left(\frac{7^{-4}}{4^{-2}}\right)}^{\frac{1}{4}}\left[\begin{array}{ccc}1& -1& 2\\ 3& 0& -2\\ 1& 0& 3\end{array}\right]$

$==\left(\frac{\frac{1}{7}}{\frac{1}{2}}\right)\left[\begin{array}{ccc}1& -1& 2\\ 3& 0& -2\\ 1& 0& 3\end{array}\right]$

$=\left(\frac{2}{7}\right)\left[\begin{array}{ccc}1& -1& 2\\ 3& 0& -2\\ 1& 0& 3\end{array}\right]$

$adjA=\frac{2}{7}adj\left[\begin{array}{ccc}1& -1& 2\\ 3& 0& -2\\ 1& 0& 3\end{array}\right]$

$=\left(\frac{2}{7}\right){\left[\begin{array}{ccc}0& -11& 0\\ 3& 1& -1\\ 2& 8& 3\end{array}\right]}^{\prime }$

$=\left(\frac{2}{7}\right)\left[\begin{array}{ccc}0& 3& 2\\ -11& 1& 8\\ 0& -1& 3\end{array}\right]$

$=\left[\begin{array}{ccc}0& \frac{6}{7}& \frac{1}{7}\\ \frac{-22}{7}& \frac{2}{7}& \frac{16}{7}\\ 0& \frac{-2}{7}& \frac{6}{7}\end{array}\right]$