given: A =

|a b c|

|d e f| det(A) = 5

|g h i|

B =

|a u c|

|d v f| det(B) = -3

|g w i|

FIND:

|-3a c 2b - u|

|-3d f 2e - v|

|-3g i 2h - w|

-it's pretty much "B" except with the column of A in it...not sure what to do about that

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- Apr 5th 2013, 03:51 PMmathlover700a question involving the determinant...
given: A =

|a b c|

|d e f| det(A) = 5

|g h i|

B =

|a u c|

|d v f| det(B) = -3

|g w i|

FIND:

|-3a c 2b - u|

|-3d f 2e - v|

|-3g i 2h - w|

-it's pretty much "B" except with the column of A in it...not sure what to do about that - Apr 9th 2013, 11:53 AMmathguy25Re: a question involving the determinant...
Given

aei + bfg + cdh - ceg - bdi - afh = 5

avi + ufg + cdw - cvg - udi - afw = -3

Find

(-3a)(f)(2h-w) + c(2e-v)(-3g) + (2b-u)(-3d)(i) - (2b - u)f(-3g) - c(-3d)(2h - w) - (-3a)(2e - v)(i) = ?

-6afh + 3afw - 6ceg + 3cvg - 6bdi + 3udi + 6bfg - 3ufg + 6cdh - 3cdw + 6aei - 3avi

[6aei + 6bfg + 6cdh - 6ceg - 6bdi - 6afh] + [-3avi - 3ufg - 3cdw + 3cvg + 3udi + 3afw]

6(aei + bfg + cdh - ceg - bdi - afh) + (-3)(avi + ufg + cdw - cvg - udi - afw)

6(5) + (-3)(-3)

30 + 9

39 - Apr 9th 2013, 03:22 PMSorobanRe: a question involving the determinant...
Hello, mathlover700!

I hope I interpreted the problem correctly . . .

Quote:

. .

. .

Multiply -3 times column-1: .

. .

Switch column-2 and column-3: .

. .

Hey, I agree with Mathguy25 ! - Apr 9th 2013, 04:22 PMmathguy25Re: a question involving the determinant...
Soroban, wow. My newbieness is showing. I attempted to approach the problem that way using the properties of manipulating columns but found no clear solution. Then I resorted to the basic way I know: brute force. Well done, sir. I'm impressed.

- Apr 9th 2013, 07:43 PMSorobanRe: a question involving the determinant...
Hello, mathguy25!

And I am impressed by your dedication and stamina

. . evident in your algebraic solution.

*Well done*to you, sir!

- Apr 9th 2013, 09:35 PMibduttRe: a question involving the determinant...
- Apr 13th 2013, 04:31 PMmathlover700Re: a question involving the determinant...
thank you mathguy25 and thank you soroban for your help! I appreciate both solutions!

- Apr 16th 2013, 01:21 PMLoblawsLawBlogRe: a question involving the determinant...
I hope I'm interpreting your solution correctly, but I think there are two errors here that cancel each other out. , where n is the dimension of A. This is because each column is multiplied by 2, and each of those multiplications multiplies the determinant by 2. In this case, .

Also, in this case*, but not in general.

What's happening here is that , where the notation means the matrix that's the same as A except the ith column is replaced with the column vector v, and represents the ith column of A, so does indeed equal 13 and the rest of the proof works fine.

*edit: whoops, no it doesn't. It would if det(2A)=2det(A) is what I should have said.