Question

For an invertiblenxnmatrix A, what is the relationship betweenadj(A)and$\mathit{a}\mathit{d}\mathit{j}\mathbf{}\left(A^{-1}\right)$ a?

Short answer

Therefore, the relationship of adj (A) and $\mathrm{adj}\left({\mathrm{A}}^{-1}\right)$is given by,

$\mathrm{adj}\left({\mathrm{A}}^{-1}\right)={\left(\mathrm{adj}\mathrm{A}\right)}^{-1}$

Step-by-step solution

 Step 1: Matrix Definition. 

Matrix is aset of numbers arranged in rows and columns so as to form a rectangulararray.

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 the relationship of adj(A) and adj(A-1) 

To find the relationship,

It applies,

$$\begin{gathered} \mathrm{adj}\left({\mathrm{A}}^{-1}\right)=\left(\det {\mathrm{A}}^{-1}\right)\mathrm{A} \\ \mathrm{adj}\left({\mathrm{A}}^{-1}\right)=\frac{1}{\det \mathrm{A}}\mathrm{A} \\ \mathrm{adj}\left({\mathrm{A}}^{-1}\right)={\left(\left(\det \mathrm{A}\right){\mathrm{A}}^{-1}\right)}^{-1} \\ \mathrm{Therefore} \\ \mathrm{adj}\left({\mathrm{A}}^{-1}\right)={\left(\mathrm{adj}\mathrm{A}\right)}^{-1} \end{gathered}$$