Strange determinant proof (how to prove)

Thread Starter

dcbingaman

Joined Jun 30, 2021
720
I am reading a book on Linear algebra. The following problem was presented in the first chapter of the book concerning Determinants:

1661796802545.png
In other words, if we consider each row in the matrix to be a number in decimal and if all the rows taken as such are divisible by k then the determinant is also divisible by k.

But how to do go about proving this?

I verified it with a simpler 2x2 matrix:

1661796533139.png

So it appears to apply to any matrix of any order. I have no idea how you go about demonstrating (proving) that?
 

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Thread Starter

dcbingaman

Joined Jun 30, 2021
720
I am reading a book on Linear algebra. The following problem was presented in the first chapter of the book concerning Determinants:

View attachment 275070
In other words, if we consider each row in the matrix to be a number in decimal and if all the rows taken as such are divisible by k then the determinant is also divisible by k.

But how to do go about proving this?

I verified it with a simpler 2x2 matrix:

View attachment 275069

So it appears to apply to any matrix of any order. I have no idea how you go about demonstrating (proving) that?
I found a solution to this which is very elegant and simple and it works for any matrix of any order:

1661806510110.png

https://math.stackexchange.com/ques...9-show-that-its-determinant-is-also-divisible
 

MrSalts

Joined Apr 2, 2020
2,350
How often has anyone used Linear Algebra OR calculus while working as an engineer. I'm guessing less than 20% of people with a BS - engineering degree (as their highest degree) will answer yes.
 

Suncalc

Joined Mar 23, 2021
11
How often has anyone used Linear Algebra OR calculus while working as an engineer. I'm guessing less than 20% of people with a BS - engineering degree (as their highest degree) will answer yes.
I for one will answer yes to using linear algebra fairly often over my 33 years as an RF systems Engineer (I used calculus weekly & sometimes almost daily). Any network stability or sensitivity analysis starts off with a state matrix of partial derivative terms which must be used in the solution of the n-dimensional state equation. This requires the use of linear algebra by finding determinants, inverting and multiplying matrices, and sometimes using L/U decompositions to set up computer solutions. Linear algebra is one of the tools of choice for analog network design and analysis.
 

Thread Starter

dcbingaman

Joined Jun 30, 2021
720
How often has anyone used Linear Algebra OR calculus while working as an engineer. I'm guessing less than 20% of people with a BS - engineering degree (as their highest degree) will answer yes.
I will have to agree it is rare. I used calculus maybe 4-5 times per year for unique problems in engineering (both software and hardware). I can't recall using Linear Algebra for any of the designs during my career except for 3D graphics programming where it came in handy a lot.
When I worked R&D for the Grinding industry (centerless grinders, cam shaft and crank shaft grinders) calculus was used a lot in the mechanical design considerations of those types of machines).
 
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Thread Starter

dcbingaman

Joined Jun 30, 2021
720
How often has anyone used Linear Algebra OR calculus while working as an engineer. I'm guessing less than 20% of people with a BS - engineering degree (as their highest degree) will answer yes.
One thing I have learned about LA. It seems a lot easier just to have a computer do the calculations for Determinants or even RREF. Considering how complicated it gets with even a modest size square matrix. To find the determinant of a 10x10 matrix requires 10! terms of 10 factors each for 3,628,800 terms! Even solving a 6 equation with 6 unknowns is insanely complicated with 720 terms! Like that would be 27 pages of calculations by hand assuming you do not make any mistakes!
 
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MrAl

Joined Jun 17, 2014
9,533
One thing I have learned about LA. It seems a lot easier just to have a computer do the calculations for Determinants or even RREF. Considering how complicated it gets with even a modest size square matrix. To find the determinant of a 10x10 matrix requires 10! terms of 10 factors each for 3,628,800 terms! Even solving a 6 equation with 6 unknowns is insanely complicated with 720 terms! Like that would be 27 pages of calculations by hand assuming you do not make any mistakes!
Hi,

I dont think anyone does this by hand anymore right?
I use math software except in some cases of really large matrices like 1000x1000 where i use my own programs.
Sometimes solutions are not due to inverting the matrix and multiplying like many are, but each 'cell' is updated over and over until the matrix becomes stable. I good example of this is solving a resistor grid. It's an interesting problem but cant do it without some computer help of some kind it would take a lifetime to solve.
 
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