Exercise 1.6.20

If the only eigenvectors of $A$ are multiples of $(1,4)$, then A has

• 1.

  no inverse: False. It’s possible that a matrix $A$ has full rank but with only one independent eigenvector. It may not have zero eigenvalue.

• 1.

  a repeated eigenvalue: True, an eigenvector is missing which can only happen when there’s repeated eigenvalue (wrong answer: False, the $\mathit{GM}=1$, so we have $\mathit{AM}\ge \mathit{GM}=1$, if we have $\mathit{AM}=1$ as well, there no repeated eigenvalue.)

• 1.

  no diagonalization $\mathit{X\Lambda }{X}^{-1}$: True, we don’t have $X^{-1}$ available.