A quiz question based on matrices over $\Bbb C$

linear algebramatrices

I am trying previous years questions of my linear algebra exam and I was unable to solve this particular question.

State True/False with reasoning: Let $A$ belong to $M_{n \times n}(\mathbb{C})$. Then:

  1. There exists a matrix $B$ belonging to $M_{n \times n}(\mathbb{C})$ such that $B^{2}=A$.
  2. There exists an invertible matrix $P$ such that $PAP^{-1}$ is upper triangular.

For 1. I could think of comparing both sides but that is feasible for only $2 \times 2$ matrices.

For 2. I have no idea on which result I should use .

Any hints would be really appreciated .

Best Answer

  1. Consider $$A= \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$$

If there exists $B$ such that $B^2=A$, then $B^4=A^2=0$, so $B$ is nilpotent, but $B$ has size $2$, so $B^2=0$, so $A=0$. This is absurd.

  1. This is very classical, see for example : https://en.wikipedia.org/wiki/Triangular_matrix#Triangularisability
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