# A determinant containing a variable

by Quark Itself
Tags: determinant, variable
 P: 25 1. The problem statement, all variables and given/known data I have this 4x4 determinant and usually these are just mechanical work, until I stumbled upon one containing x, how should one go about solving these type of determinants? [x 2x 4 x ] [1 2 2x 1 ] [2x x-1 2 3x] [ 2 x+1 x+3 x-1] What are the different values of x? 2. Relevant equations 3. The attempt at a solution I've tried ellimination and by using cofactors. When fully computed, it will be a polynomial of the 4th degree, as one probably can see. Ellimination didn't make it simpler to factorize and cofactors just solved it until the polynomial popped up at the end-
Mentor
P: 21,412
 Quote by Quark Itself 1. The problem statement, all variables and given/known data I have this 4x4 determinant and usually these are just mechanical work, until I stumbled upon one containing x, how should one go about solving these type of determinants? [x 2x 4 x ] [1 2 2x 1 ] [2x x-1 2 3x] [ 2 x+1 x+3 x-1] What are the different values of x? 2. Relevant equations 3. The attempt at a solution I've tried ellimination and by using cofactors. When fully computed, it will be a polynomial of the 4th degree, as one probably can see. Ellimination didn't make it simpler to factorize and cofactors just solved it until the polynomial popped up at the end-
What did you get?
 P: 25 8x4-22x3-22x2+47x+3
HW Helper
Thanks
PF Gold
P: 7,721
A determinant containing a variable

 Quote by Mark44 What did you get?
 Quote by Quark Itself 1. The problem statement, all variables and given/known data I have this 4x4 determinant and usually these are just mechanical work, until I stumbled upon one containing x, how should one go about solving these type of determinants? [x 2x 4 x ] [1 2 2x 1 ] [2x x-1 2 3x] [ 2 x+1 x+3 x-1] What are the different values of x?
 Quote by Quark Itself 8x4-22x3-22x2+47x+3
Is this your determinant?$$\left|\begin{array}{cccc} x & 2x & 4 & x \\ 1 & 2 & 2x & 1\\ 2x & x-1 & 2 & 3x\\ 2 & x+1 & x+3 & x-1 \end{array}\right|$$
If so, your last two terms should be ##44x+12##. Still, once you have the polynomial correct, you haven't said what you are supposed to do with it. It doesn't make sense to ask "what are the different values of ##x##".
Math
Emeritus
Thanks
PF Gold
P: 39,691
 Quote by Quark Itself 1. The problem statement, all variables and given/known data I have this 4x4 determinant and usually these are just mechanical work, until I stumbled upon one containing x, how should one go about solving these type of determinants? [x 2x 4 x ] [1 2 2x 1 ] [2x x-1 2 3x] [ 2 x+1 x+3 x-1] What are the different values of x?
The different values of x that give the determinant what property? That it be 0? That it not be 0?

 2. Relevant equations 3. The attempt at a solution I've tried ellimination and by using cofactors. When fully computed, it will be a polynomial of the 4th degree, as one probably can see. Ellimination didn't make it simpler to factorize and cofactors just solved it until the polynomial popped up at the end-

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