Properties of the Determinant II

Algebra Level 3

Some matricies, like the Vandermonde matrix , have a determinant with a cute factorisation.

If $\begin{vmatrix} 1 & a_1 & a_1^2 \\ 1 & a_2 & a_2^2 \\ 1 & a_3 & a_3^2\end{vmatrix} \neq 0$ , what can we say about $a_1$ , $a_2$ , and $a_3$ ?

a_1

a_2

, and

a_3

are positive

a_1

a_2

, and

a_3

are equal

a_1

a_2

, and

a_3

sum to zero

a_1

a_2

, and

a_3

are distinct

a_1

a_2

, and

a_3

are nonzero

1 solution

Adam K
Apr 23, 2016

We know the factorization of the determinant of the Vandermonde matrix of size $n \times n$ is given by the product:

$\prod_{1\leq i\lt j \leq n}(\lambda_j - \lambda_i)$

Since this has $n=3$ , then the determinant is $(a_2-a_1)(a_3-a_1)(a_3-a_2)$ . (There are other ways to find this, but the factorization is hard). If this is to be nonzero, then none of $a_2-a_1$ , $a_3-a_1$ or $a_3-a_2$ can be zero. Therefore $a_1$ , $a_2$ and $a_3$ are distinct.

Nice solution! (+1) Small typo: It should be $i<j$ in the product, of course.

Otto Bretscher - 5 years, 1 month ago

Properties of the Determinant II

1 solution

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