Question

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.
\[
\underline{a}\times (\underline{b}\times \underline{c}) = \underline{b}(\underline{a}\cdot\underline{c}) - \underline{c}(\underline{a}\cdot\underline{b})
\]

Answer 1

for first question try below :
R2 - R1
R3 - R1

Answer 2

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!

Answer 3

Proven by standard Gaussian elimination. I was expecting some ingenious prove without using it.

Answer 4

*proofs