Question

Show that matrix 0 0 1 0 1 0 1 0 0 is invertible find its inverse by adjoint method

Answer 1

$\underline{\textbf{Given:}}$

$\left(\begin{array}{ccc}0&0&1\\0&1&0\\1&0&0\end{array}\right)$

$\underline{\textbf{To prove:}}$

$\left(\begin{array}{ccc}0&0&1\\0&1&0\\1&0&0\end{array}\right)\;\textsf{is invertible}$

$\underline{\textbf{Solution:}}$

$A=\left(\begin{array}{ccc}0&0&1\\0&1&0\\1&0&0\end{array}\right)$

$|A|=\left|\begin{array}{ccc}0&0&1\\0&1&0\\1&0&0\end{array}\right|$

$\mathsf{|A|=0(0-0)-0(0-0)+1(0-1)}$

$\mathsf{|A|=-1}$

$\implies\mathsf{|A|\;\neq\;0}$

$\therefore\textbf{A is invertible}$

$\mathsf{Adjoint\;of\;A=(Cofactor\;matrix\;of\;A)^T}$

$\implies\mathsf{adj\,A=\left(\begin{array}{ccc}+(0-0)&-(0-0)&+(0-1)\\-(0-0)&+(0-1)&-(0-0)\\+(0-1)&-(0-0)&+(0-0)\end{array}\right)^T}$

$\implies\mathsf{adj\,A=\left(\begin{array}{ccc}0&0&-1\\0&-1&0\\-1&0&0\end{array}\right)^T}$

$\implies\mathsf{adj\,A=\left(\begin{array}{ccc}0&0&-1\\0&-1&0\\-1&0&0\end{array}\right)}$

$\mathsf{Now,}$

$\bf\,A^{-1}=\dfrac{1}{|A|}\,adj\,A$

$A^{-1}=\dfrac{1}{-1}\left(\begin{array}{ccc}0&0&-1\\0&-1&0\\-1&0&0\end{array}\right)$

$A^{-1}=-1\left(\begin{array}{ccc}0&0&-1\\0&-1&0\\-1&0&0\end{array}\right)$

$\boxed{\bf\,A^{-1}=\left(\begin{array}{ccc}0&0&1\\0&1&0\\1&0&0\end{array}\right)}$