Is the matrix 1101 diagonalisable?
$$ \newcommand\onemat{ \begin{pmatrix} 1 & 1 \\ 0 & 1 \\ \end{pmatrix} } $$ Is the matrix $\onemat$ diagonalisable? To be diagonalisable, it requires that the matrix $M$ is similar to a diagonal matrix $D$, in other words $D=A^{-1}MA$ for some invertible matrix $A$. Then $$ \begin{align} \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} \onemat \frac1{ad-bc} \begin{pmatrix} d & -b \\ -c & a \\ \end{pmatrix} &= \begin{pmatrix} ad-ac-ad & a^2\\ -c^2 & ad+ac-bc\\ \end{pmatrix} \end{align} $$For this to be diagonal, $a^2=c^2=0$, so $a=c=0$, but then both of the diagonal entries are zero, so the matrix is not diagonalisable.
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