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I know that any nilpotent matrix $M$ has $\det(M)=0$, because $M^k=0$ and thus $\det(M^k)=0$. Are there any simple examples of matrices $A$ that do have $\det(A)=0$ that are not nilpotent? I've tried to find one myself, but I couldn't find one. A proof that there is such a matrix would suffice if it is pretty simple/intuitive.

Ragnar
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  • Take the identity on $\mathbb{R^{n}}$ for any $n > 1$, and set one of the 1's on the diagonal to 0. The resulting matrix $M$ has 0 determinant and is not nilpotent because $M^{k}=M\ne 0$ for all $k \ge 1$. – Disintegrating By Parts Feb 25 '14 at 20:47

2 Answers2

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Consider $A=\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$. Clearly $A^k=A \neq 0$, but $\det(A)=0$.

Jason
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The matrix $\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$.

John Hughes
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