Matrix Determinant with Added Rows: How Does Multiplication Affect Calculation?

[nicknaq]

Homework Statement

matrix A= [v1,v2,v3,v4]^T and detA=4
then what is det [9v1+7v4, v2, v3, 9v4+4v1]^T ?

Homework Equations

None.

The Attempt at a Solution

I know that my problem is identical to the problem encountered in https://www.physicsforums.com/showthread.php?t=431395"
However, I don't know how the addition of 7v4 and 4v1 (for rows 1 and 4, respectively) will alter the calculation. I know that adding 4v1 is different than usual because row 1 has been multiplied by 9. Similarly, adding 7v4 is different because row 4 has been multiplied by 9. But what happens first? And how does it affect the det calculation?

Thank you,
Nick

 

[hgfalling]

This is a straightforward application of the determinant rules. There's a summary of those here:

http://en.wikipedia.org/wiki/Determinant#Properties_characterizing_the_determinant

 

[HallsofIvy]

If you multiply one row (or column) of a determinant by a number, the entire determinant is multiplied by that number. If you add a multiple of one row (or column) to another, the determinant stays the same. (If you swap two rows (or columns) you multiply the determinant by -1 but that does not happen here.)

 

[nicknaq]

I understand the rules. I'm having troubles applying them.

Here's what I gather:
v1 : 9v1+7v4 --> 9v1+7*(4v1+9v4) = 37v1 + 63v4
v4 : 4v1+9v4--> 9v4+4*(9v1+7v4)= 37v4 + 36v1

Thus the det should be 4*37*37. But that is incorrect. Where is my error?

 

[crayzwalz]

yo ur that guy from the warums lol

 

[nicknaq]

huh?

 

[Dick]

It's the same answer as in the other thread. det[9v1+7v4, v2, v3, 9v4+4v1]=det[9v1,v2,v3,9v4+4v1]+det[7v4,v2,v3,9v4+4v1]. Now expand in the fourth row or column or whatever.

 

[nicknaq]

It's the same answer as in the other thread. det[9v1+7v4, v2, v3, 9v4+4v1]=det[9v1,v2,v3,9v4+4v1]+det[7v4,v2,v3,9v4+4v1]. Now expand in the fourth row or column or whatever.

sorry but I'm still not getting it.

given those matrices, my det should be (9*9*4)+(9*7*4), no?

 

[Dick]

sorry but I'm still not getting it.

given those matrices, my det should be (9*9*4)+(9*7*4), no?

No. If you expand the second two you'll get 4 dets. Two of them will be zero. Two of them will be nonzero. Which ones are nonzero and what's the value of each one?

 

[nicknaq]

No. If you expand the second two you'll get 4 dets. Two of them will be zero. Two of them will be nonzero. Which ones are nonzero and what's the value of each one?

What do you mean by "expand" the second matrix?

 

[Dick]

What do you mean by "expand" the second matrix?

Determinants are linear. E.g. det(a+b,c,d,e)=det(a,c,d,e)+det(b,c,d,e). That's what I mean by "expand".

 

[nicknaq]

Ugh. I give up.

Thanks for trying. My fault, not yours.

 

[Dick]

Ugh. I give up.

Thanks for trying. My fault, not yours.

S'ok, but you are giving up on a pretty easy problem, really. If you want to give it another shot did you follow what I did in post 8 given the rule I gave in post 12? If you just want to pack it in, that's ok too.