Given 4 2x2 real matrices, which one is not diagonalisable?
A. $\begin{pmatrix} 1 & 0\\ 3 & 1 \end{pmatrix}$ B. $\begin{pmatrix} 2 & 1\\ 0 & 3 \end{pmatrix}$ C. $\begin{pmatrix} 2 & 0\\ 1 & 3 \end{pmatrix}$ D. $\begin{pmatrix} 3 & -1\\ -1 & 3 \end{pmatrix}$
I think it is A. because the algebraic multiplicity there is $2$ but the only eigenvector is of the form $\begin{pmatrix} \alpha\\ 0 \end{pmatrix}$, so the geometric multiplicity is $1$ thus $\text{AM}=2\neq 1=\text{GM} \implies \text{not diagonalisable}$
Is this correct?
Thanks.
The eigenvector is actually of the form $\begin{pmatrix} 0 \\ \alpha\end{pmatrix}$, otherwise you are correct.
The author of the problem clearly doesn't want you to do any computation at all. (B) and (C) have distinct eigenvalues, hence must be diagonalisable. (D) is a real symmetric matrix. So the only possibile answer is (A).