Is there a specific way to find out if a...

If images are not shown try to refresh the page. If you like this website, please disable any AdBlock software!

You are currently reading a thread in /sci/ - Science & Math

You are currently reading a thread in /sci/ - Science & Math

Thread images: 1

Is there a specific way to find out if a matrix's rank depends on a variable inside said matrix?

In this example I've just been trying to replace beta (alpha being any value) with random values until I get the rank down to 2.

>>

Use GauB

>>

>>7767966

Even after Gauss it's hard to get to a conclusion, unless I'm doing it wrong.

>>

>>7767964

The rank of a matrix is equal to the size of the biggest non zero determinant you can calculate from it.

That matrix already has got at least two as rank, since it has a non zero 2x2 size determinant. To test if its rank is three, simply calculate the bigger determinant possible, which is the unique of size 3x3.

>>

>>7767974

Beta doesn't matter. The rank is 2 if alpha is 1 and 3 for all other values of alpha.

>>

>>7767988

I don't understand how can a matrix have different determinants, are you talking about cofactors, like the 2x2 determinants inside a 3x3 matrix?

>>

>>7767990

Why doesn't beta matter? how did you get to that conclusion?

>>

>>7767995

Call it as you like, but its accurate name is minor.

https://en.wikipedia.org/wiki/Minor_%28linear_algebra%29

>>

>>7767995

obviously he is. I didn't know those were called cofactors in english though, in my country we call them subdeterminants or minors

>>

>>7767990

No, the rank is 3 for α≠1. For α=1 it is 2.

>>

>>7767964

can the matrix have rank 1?

No because the first two columns are linearily independant whatever the value of beta is.

So the rank of your matrix is either 2 or 3.

Now just compute the determinant as a function of alpha and beta and find the values for which the det. is equal to 0 or different from 0. That will give you the cases where rank(A) is 2 and rank(A) is 3

>>

>>7768000

[eqn] \begin{bmatrix} -1 & 0 & 0 \\ 1 & 1 & \alpha \\ \beta & 1 & 1 \end{bmatrix} [/eqn]

Add the first row to the second row and add beta times the first row to the second row.

[eqn] \begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & \alpha \\ 0 & 1 & 1 \end{bmatrix} [/eqn]

Subtract the second row from the last row.

[eqn] \begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & \alpha \\ 0 & 0 & 1 - \alpha \end{bmatrix} [/eqn]

It's triangular iff alpha = 1.

>>

>>7767988

cool, I didn't know that was the case. thanks

>>

>>7768013

No, it is already triangular whatever alpha is.

>>

>>7768013

>It's triangular iff alpha = 1.

you mean singular

Thread images: 1

Thread DB ID: 377935

All trademarks and copyrights on this page are owned by their respective parties. Images uploaded are the responsibility of the Poster. Comments are owned by the Poster.

This is a 4chan archive - all of the shown content originated from that site. This means that 4Archive shows their content, archived. If you need information for a Poster - contact them.

If a post contains personal/copyrighted/illegal content, then use the post's