Question

For an invertiblenxnmatrix A, what is adj(adj A)?

Short answer

Therefore, the adj (adj A) is given by,

$adj(adjA)={(detA)}^{n-2}A$

Step-by-step solution

Matrix Definition. 

Matrix is a set of numbers arranged in rows and columns so as to form a rectangular array.

The numbers are called the elements, or entries, of the matrix.

If there are m rows and n columns, the matrix is said to be an “m by n” matrix, written “.m x n”

To find

To find adj(adj A),

It applies,

$$\begin{gathered} adj\left(adjA\right)=adj\left(\left(detA\right)A^{-1}\right) \\ adj\left(adjA\right)=\left(det\left(\left(detA\right)A^{-1}\right){\left(\left(detA\right)A^{-1}\right)}^{-1}\right) \\ adj\left(adjA\right)={\left(detA\right)}^{n}det{A}^{-1}.\frac{1}{detA}A \\ adj\left(adjA\right)={\left(detA\right)}^{n-2}A \\ \mathrm{Therefore}, \\ \mathrm{adj}\left(\mathrm{adj}\mathrm{A}\right)={\left(\det \mathrm{A}\right)}^{\mathrm{n}-2}\mathrm{A}. \end{gathered}$$