<s>[INST] To find jordan canonical form Which of the following matrices have Jordan canonical form of equal to the $3\times 3$ matrix  
$$
\begin{pmatrix}
0 & 1 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}$$
a)$
\begin{pmatrix}
0 & 0 & 1 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}$
b)$
\begin{pmatrix}
0 & 0 & 1 \\
0 & 0 & 1 \\
0 & 0 & 0
\end{pmatrix}$
c)$
\begin{pmatrix}
0 & 1 & 1 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}$
d)$
\begin{pmatrix}
0 & 1 & 1 \\
0 & 0 & 1 \\
0 & 0 & 0
\end{pmatrix}$
Here characteristic equation of the matrix is $x^3$.Hence the 3 eigenvalues of the matrix are zero.
Do we want to find the eigenvalues of all the matrices in the options?Is there any other way?
 [/INST] Finding the eigenvalues of any of these matrices is not particularly difficult; they are all upper-triangular.
Hint: Suppose that $A$ is a $3 \times 3$ matrix with zero as its only (complex) eigenvalue.  Note that $A$ has the desired J-C form if and only if $A \neq 0$ but $A^2 = 0$ (why?).
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