If $A \in L(\mathbb {R}^n)$ with $\Vert A \Vert _{{op}} \neq 0$, then $A \in GL_n$.

Question

Which of these following statements is true? $n \in \mathbb {N}$

a) It exists $A, B \in L(\mathbb {R}^n)$ with $\Vert A\Vert _{{op}} \neq 0$, $\Vert B\Vert _{{op}} \neq 0$, but $\Vert AB\Vert _{{op}} = 0$.

b) If $A \in L(\mathbb {R}^n)$ with $\Vert A \Vert _{{op}} \neq 0$, then $A \in GL_n$.

I think that a) is false and b) is true, is that correct?

Answer 1

Remember that linear operators on $\mathbb R^n$ can be represented by matrices. For $a)$ just find a non-zero nilpotent matrix. For example $$A=\begin{pmatrix}0 & 1\\ 0 & 0\end{pmatrix}.$$ $b)$ is just asking whether every nonzero matrix is invertible, which is obviously false.