If the Rank of matrix A is n-1 then there is atleast one minor present of order n-1 of the matrix A is not equal to zero.Therefore the matrix Adj A will be a non zero matrix and thus the Rank of the matrix Adj A will be greater than zero.
Now
Rank of matrix A is n-1 Therefore | A | =0
Therefore A ( Adj A) will be a Zero matrix.
and we know
If the Rank of matrix A is n-1 then there is atleast one minor present of order n-1 of the matrix A is not equal to zero.
Hence we can say directly Adj A will not be Zero.