2016 UoTokyo Mathematics (An Error Report)

[ttthinker]

In the problem of Linear Algebra (Problem 1), the solution of (3) subproblem is a little error.
After row transformation of $(A-\lambda E)$, the matrix will be:
$$
\begin{pmatrix}
1 & -\lambda & 0 \
0 & 1 & -\lambda \
0 & 0 & P(\lambda)
\end{pmatrix}
$$
, where $P(\lambda)$ is the characteristic polynomial of $A$.
Thus, $P(\lambda) = 0$ and the equation $(A-\lambda E) x_k = 0$ has non-zero solution $x_k = (\lambda_k^2, \lambda_k, 1)$