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I will denote by $\operatorname{adj}(A)$ the adjugate matrix of $A$ i.e. the transpose of the cofactor matrix,which has the propery that $A\cdot \operatorname{adj}(A)=\operatorname{adj}(A)\cdot A=\det A \cdot I_n$.
I am curious to find nonzero matrices(no matter the dimension) whose adjugate is the zero matrix. I could only come up with examples such as $A=\begin{pmatrix} 1 & 1 & 1 \\ 2 & 2 & 2 \\ 3 & 3 & 3 \\ \end{pmatrix}$,but I want to see some other ones.

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    $\begingroup$ another example: $\left(\begin{matrix}1 & 2 & 3 \\ 1 & 2 & 3 \\ 2 & 4 & 6 \end{matrix}\right)$ $\endgroup$ – J. W. Tanner Apr 15 at 20:26
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    $\begingroup$ @Math Guy: If $A$ is an $n\times n$-matri with $rk(A) < n-1$, then $adj(A) = 0$, that is, $adj(A)$ is the zero matrix; if $rk(A) = n-1$, then $rk(adj(A)) = 1$; if $rk(A) = n$, then $rk(adj(A)) = n$. That's it. $\endgroup$ – Moritz Apr 15 at 20:27
  • $\begingroup$ @Moritz that's right, I hadn't thought about using this! $\endgroup$ – Math Guy Apr 15 at 20:53
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Any matrix of the form$$\begin{bmatrix}a&b&c\\\alpha a&\alpha b&\alpha c\\\beta a&\beta b&\beta c\end{bmatrix}$$will do.

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