Exercise 3.3.5

Question

A square matrix is called $nilpotent$ if $A^k = \mathbf{0}$ for some positive integer $k$. Show that for a nilpotent matrix $A, \, \det A = 0$.

Jing Huang · Accepted Answer

\begin{proof}
$\det A^k = (\det A)^k = \det \mathbf{0} = 0$, thus $\det A = 0$.    
\end{proof}

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