Exploring the Role of Determinants in Orienting Scientific Research

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.
  • #1
Lambda96
171
63
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

Bildschirmfoto 2023-05-10 um 15.01.52.png

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.

Bildschirmfoto 2023-05-10 um 15.28.28.png
 
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  • #2
I hope you can read my calculation well, otherwise I will upload the calculation again in a higher resolution
 
  • #3
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 [itex]\mathbf{w}_n = D_1\mathbf{e}_1 + \dots + D_{n}\mathbf{e}_{n}[/itex].
 
Last edited:
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  • #4
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.
 
  • #5
This is a typo, right?
1683777409904.png

should be ##w_1^2##
 
  • #6
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 :smile:
 
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