If a matrix is non-zero then is it possible that Adj A is zero matrix ?

Question

For this matrix although A is non-zero but adjoint matrix is zero , so is it true that Adj of a matrix need not be always non-zero for a non-zero matrix ?

Answer 1

Yes,It is true. Adj of a non zero matrix need not be non-zero or Zero always.

A= $\begin{bmatrix} 1 & 1 & 1\\ 1& 1& 1\\ 1& 1 & 1 \end{bmatrix}$

For this example A is non Zero but Adj A is Zero

Take one more Example

B= $\begin{bmatrix} 1 & 2\\ 3& 4 \end{bmatrix}$

For this Example B is non Zero and Adj B is Non Zero

Hence,  Adj of a non zero matrix need not be non-zero or Zero always.