# Exploring the Role of Determinants in Orienting Scientific Research

• Lambda96
In summary, the conversation discusses a problem with a task and the use of the Laplace evolution theorem and determinant to calculate ##W_n##. The process involves forming a scalar product ##W_1 W_n##, but when reversing the process, the unit vectors disappear and cause a typo in the determinant. The mistake is noted and will be corrected.
Lambda96
Homework Statement
Show that the vector ##W_n## is orthogonal to all vectors ##w_j##
Relevant Equations
none
Hi,

unfortunately, I have problems with the task c

I used the tip with the Laplace evolution theorem and rewrote the determinant to calculate ##W_n##. Then I simply formed the scalar product ##W_1 W_n## and here I get now no further.

I hope you can read my calculation well, otherwise I will upload the calculation again in a higher resolution

To get to this point, you expanded a determinant by its final column. What happens if you reverse that process after calculating the dot product?

Hint: Write $\mathbf{w}_n = D_1\mathbf{e}_1 + \dots + D_{n}\mathbf{e}_{n}$.

Last edited:
Lambda96 and malawi_glenn
Thanks pasmith for your help and the tip

I have now calculated the scalar product with the ##\vec{W_1}##, so ##\vec{W_1} \cdot \vec{W_n}## Now if I reverse the process again, that is the Laplace expansion then the unit vectors ##\vec{e_i}## disappear due to the scalar product and the column with the unit vectors in the determinant is replaced by the vector ##\vec{W_1}##, so I get the following determinant.

$$det \left( \begin{array}{rrrrr} W_1^1 & W_2^1 & \cdots & W_{n-1}^1 & W_1^1 \\ W_1^2 & W_2^2 & \cdots & W_{n-1}^2 & W_1^2 \\ \vdots & \vdots & \cdots & \vdots & \vdots \\ W_1^n & W_2^n & \cdots & W_{n-1}^n & W_1^n\\ \end{array}\right)$$

The first column now occurs twice, making the determinant zero.

This is a typo, right?

should be ##w_1^2##

You are right malawi_glenn, it is a spelling mistake, it should be ##W_1^2##, thanks for pointing it out, I will tell my lecturer so he can correct it

malawi_glenn

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