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I am not able give an example for the following problem on simultaneous triangularization. So, I thought I will post it here.

Give an example of three linear transformations $A,B$ and $C,$ such that the pairs $\lbrace A,B\rbrace$, $\lbrace B,C\rbrace$ and $\lbrace A,C\rbrace$ are simultaneously triangularizable, but the triplet $\lbrace A,B,C\rbrace$ is not simultaneously triangularizable.

Thank you.

ADDED LATER: For my need, I am looking for an example of linear transformations acting on vector spaces over $\mathbb{C}.$

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  • $\begingroup$ Silly extension: what abstract simplicial complexes can be realized with matrices corresponding to vertices and sets of simultaneously triangularizable matrices as simplices? $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Mar 7, 2012 at 16:37

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Here is a simple example: $$ A = \begin{pmatrix} 0 & 1 & \\ & 0 & \\ & & 0 \end{pmatrix}, \quad B = \begin{pmatrix} 0 & & \\ & 0 & 1 \\ & & 0 \end{pmatrix}, \quad C = \begin{pmatrix} 0 & & \\ & 0 & \\ 1 & & 0 \end{pmatrix} . $$ Every pair can be triangularized (by a permutation matrix, by the way), but $A$, $B$ and $C$ have no common eigenvector, and so these three matrices can't be simultaneously triangularized.

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  • $\begingroup$ Nailed it with such a simple answer! Thanks. $\endgroup$
    – Uday
    Commented Mar 7, 2012 at 16:57

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