If $A$ is an $n$ square matrix of rank $n-1$ then show that $adjA \neq 0$
My attempt: Since $A$ is an $n$-rowed square matrix of rank $n-1$,therefore at least one $n-1$ rowed minor of the matrix $A$ is not equal to zero. But every $n-1$ square minor of matrix $A$ is equal to in magnitude to the cofactor of some element in $|A|$. Hence at least one element in $|A|$ has its cofactor non zero. Therefore at least one element of $adj.A$ is not equal to zero. Hence $adj.A \neq O$. Please provide me alternate proof of above problem. Thanks.