Let $X$ be the matrix whose columns are eigenvectors of the matrix $A$. If the eigenvectors of $A$ or column vectors of $X$ are linearly independent, then
$(i)$ $A$ is invertible
$(ii)$ $A$ is diagonalisable
$(iii)$ $X$ is invertible
$(iv)$ $X$ is diagonalisable.
Answer:
Since the eigenvectors of $A$ are linearly independent, $A$ is diagonalisable.
i.e., $A=XDX^{-1}$, where $D$ is the diagonal matrix of eigen values of $A$.
So $(ii)$ is true.
Also since the eigenvectors of $A$ are linearly independent , the matrix $A$ is invertible.
Hence $(i)$ is also true.
But how to decide about the option $(iii)$ and $(iv)$?
Help me
Best Answer
HINT
Note that for $(iii)$ $X$ is also full rank.
For $(iv)$ it seems we do not have sufficient information.