why is this matrix rank deficient?

Question

I assumed if column vectors (and/or row vectors) are independent, then a matrix is full rank; octave tells me this matrix has rank 2. Why?

$$ \begin{matrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \\ \end{matrix} $$

Answer 1

HINT: Naming your columns $x_1,x_2,x_3$. Compute $$x_1 - 2x_2 + x_3$$

Answer 2

since your matrix is symmetric you can make your matrix congruent to a diagonal matrix.

$$ Q^T D Q = H $$ $$\left( \begin{array}{rrr} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \\ \end{array} \right) \left( \begin{array}{rrr} 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & 0 \\ \end{array} \right) \left( \begin{array}{rrr} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrr} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \\ \end{array} \right) $$