Question

Question: 13. Show that if A is invertible, then adj A is invertible, and \({\left( {adj\,A} \right)^{ - {\bf{1}}}} = \frac{{\bf{1}}}{{detA}}A\).

Short answer

Hence, adj Ais invertible and \({\left( {{\rm{adj}}\,A} \right)^{ - 1}} = \frac{1}{{\det A}}A\).

Step-by-step solution

Use the definition of invertible

Given, A is invertible. Therefore,

\(A{A^{ - 1}} = {A^{ - 1}}A = I\).

Also, \({A^{ - 1}}\) is invertible and \({\left( {{A^{ - 1}}} \right)^{ - 1}} = A\).

Use the inverse formula

By inverse formula,

\({A^{ - 1}} = \frac{1}{{\det A}}{\rm{adj}}\,A\)

This implies \({\rm{adj}}\,A\) is also invertible.

 Perform the substitution

\(\begin{array}{c}{\left( {{A^{ - 1}}} \right)^{ - 1}} = A\\{\left( {\frac{1}{{\det A}}{\rm{adj}}\,A} \right)^{ - 1}} = A\\\det A{\left( {{\rm{adj}}\,A} \right)^{ - 1}} = A\\{\left( {{\rm{adj}}\,A} \right)^{ - 1}} = \frac{1}{{\det A}}A\end{array}\)