Two proofs needed: 1. Prove or provide a counterexample that matrices of the following form must be singular. \[ \begin{pmatrix} n&n+1&n+2\\ n+3&n+4&n+5\\ n+6&n+7&n+8 \end{pmatrix} \] 2. Prove the triple product identity. \[ \underline{a}\times (\underline{b}\times \underline{c}) = \underline{b}(\underline{a}\cdot\underline{c}) - \underline{c}(\underline{a}\cdot\underline{b}) \]

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Two proofs needed: 1. Prove or provide a counterexample that matrices of the following form must be singular. \[ \begin{pmatrix} n&n+1&n+2\\ n+3&n+4&n+5\\ n+6&n+7&n+8 \end{pmatrix} \] 2. Prove the triple product identity. \[ \underline{a}\times (\underline{b}\times \underline{c}) = \underline{b}(\underline{a}\cdot\underline{c}) - \underline{c}(\underline{a}\cdot\underline{b}) \]

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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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for first question try below : R2 - R1 R3 - R1
It is very unfortunate that the lecutrer try to use the vectors \[ \underline{a}=\begin{pmatrix}1&2&3\end{pmatrix}^T\\ \underline{b}=\begin{pmatrix}4&5&6\end{pmatrix}^T\\ \underline{c}=\begin{pmatrix}7&8&9\end{pmatrix}^T \] to demonstrate the scalar triple product. He wanted to demonstrate that the scalar triple product corresponded to the volume of the parallelepiped of formed by that three vectors. As \[ \underline{a}\cdot(\underline{b}\times\underline{c})= \begin{vmatrix} 1&2&3\\ 4&5&6\\ 7&8&9 \end{vmatrix} \] and matrix of the form \[ \begin{pmatrix} n&n+1&n+2\\ n+3&n+4&n+5\\ n+6&n+7&n+8 \end{pmatrix} \] are always singular, you can see how this goes!
Proven by standard Gaussian elimination. I was expecting some ingenious prove without using it.

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*proofs

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