Question

For two invertible nxnmatrices A and B , what is the relationship between $\mathbf{adj}\left(A\right)\mathbf{,}\mathbf{adj}\left(B\right)\mathbf{,}\mathbf{}\mathbf{and}\mathbf{}\mathit{a}\mathit{d}\mathit{j}\left(AB\right)$?

Short answer

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

$\mathrm{adj}\left(\mathrm{AB}\right)=\mathrm{adj}\left(\mathrm{B}\right)\mathrm{adj}\left(\mathrm{A}\right).$

Step-by-step solution

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 “.mxn”

To find the relationship of  and

To find the relationship,

It applies,

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