Properties of matrix determinant: Find the det(u+2v-3w,w,2u) without calculating the vector u +2v-3w.

[Qwertyuiop[]]

3 vectors u,v ,w , find the det(u+2v-3w,w,2u) without calculating the vector u +2v-3w. Not sure how to do this, I think I have to use some property of determinants to find the determinant without calculating the vector u+2v-3w.

 

[blamocur]

3 vectors u,v ,w , find the det(u+2v-3w,w,2u) without calculating the vector u +2v-3w. Not sure how to do this, I think I have to use some property of determinants to find the determinant without calculating the vector u+2v-3w.

What else is given? E.g., is [imath]\det (u,v,w)[/imath] known?

 

[Qwertyuiop[]]

What else is given? E.g., is [imath]\det (u,v,w)[/imath] known?

yes, i found the determinant in the previous question.

 

[blamocur]

Cannot you use multilinearity of determinants here?

 

[Qwertyuiop[]]

Cannot you use multilinearity of determinants here?

I think yes. I write det(u+2v-3w,w,2u) as det(u,w,2u) + det(2v,w,2u) + det(-3w,w,2u). factoring out the constants , 2 det(u,w,u) + 4 det(v,w,u) -6 det(w,w,u). Is that correct? Also if we have 2 identical rows, determinant is 0 right so that is just equal to 4 det(u,v,w)?

 

[blamocur]

I think yes. I write det(u+2v-3w,w,2u) as det(u,w,2u) + det(2v,w,2u) + det(-3w,w,2u). factoring out the constants , 2 det(u,w,u) + 4 det(v,w,u) -6 det(w,w,u). Is that correct? Also if we have 2 identical rows, determinant is 0 right so that is just equal to 4 det(u,v,w)?

Looks good to me.

 

[Steven G]

You can get any multiply of w in the 2nd entry that you want and any multiply of u that you want in the 3rd entry. You can use that to give off the 2v and -3w in the 1st entry leaving you with det(2v, w, 2u) = 4det(v, w, u)

 

[khansaheb]

multiply

and

multiply

By "multiply" did you mean "multiple"?