- #1
grassstrip1
- 11
- 0
The problem I have is this: Show that
\begin{bmatrix} 1 & 1 & 1 \\ λ_{1} & λ_{2} & λ_{3} \\ λ_{1}^{2} & λ_{2}^{2} & λ_{3}^{2} \end{bmatrix}
Has determinant
$$ (λ_{3} - λ_{2}) (λ_{3} - λ_{1}) (λ_{2} - λ_{1}) $$
And generalize to the NxN case (proof not needed)Obviously solving the 3x3 was not hard, I simply expanded the expression for the determinate given and showed it to be the same as the one i calculated using the rule of Sarrus.
However for the nxn case I'm not sure how to proceed. I tried entering bigger and bigger matrices into wolfram but there was no clear pattern. Is there a non computational way to solve this?
\begin{bmatrix} 1 & 1 & 1 \\ λ_{1} & λ_{2} & λ_{3} \\ λ_{1}^{2} & λ_{2}^{2} & λ_{3}^{2} \end{bmatrix}
Has determinant
$$ (λ_{3} - λ_{2}) (λ_{3} - λ_{1}) (λ_{2} - λ_{1}) $$
And generalize to the NxN case (proof not needed)Obviously solving the 3x3 was not hard, I simply expanded the expression for the determinate given and showed it to be the same as the one i calculated using the rule of Sarrus.
However for the nxn case I'm not sure how to proceed. I tried entering bigger and bigger matrices into wolfram but there was no clear pattern. Is there a non computational way to solve this?