How does this determinant equal zero?

Factual Questions
1

(Note: This isn’t homework; I’m just trying to improve my math skills with an old algebra book that I have.)

Here’s the determinant: [ul]
[li]10 8 -2[]2 -1 2[]-3 -5 3[/ul][/li]
and I’m to show that it equals zero without doing the expansion. I know that if two rows are identical, or two columns, it is equal to zero, as it is if all the elements of one row or column are zero.

I also know that if I can factor one of the rows or columns so that the common factor goes outside and the resulting row or column is equal to another, then that would show that the determininant is equal to zero. But I can’t see any appropriate factorization. What other methods are there, or am I just not seeing it?

2

IIRC you are “allowed” to add or substract multiples of one row away from another. If doing this you manage to create a row of column full of zeros than the determinant is zero.

In this case take 2R2 from R1
add 2R3 to R1

et voila.

3

Of course, I meant “row OR column”, but Mr Moderator – can I not edit my own posts? I’m refused permission – is this some feature that has been disabled?

4

And of course that should say Mr/Mrs/Ms/Dr/Rev/Bishop/Professor/PC/WPC/etc Moderator

5

To expand upon what The Great Unwashed said, if the rows/columns of a matrix are not linearly independent (i.e., if one of the rows/columns can be expressed as a linear combination of the other rows/columns), then the determinant is zero.

btw, here’s a matrix whose first column is the sum of the rows:



2  1  -1
3  1  -2
-3 1   0

You’ll note that it has determinant four (assuming I didn’t screw up), so you need to match rows with rows and columns in the explanation above.

6

Thanks, TGW, that’s the part I forgot.

Thanks to everyone for the replies so far. I’ve noticed that math questions on the SDMB almost always garner many replies.

7

By the way, congratulations on your third millenial post.

I’m not sure I follow you here. In the first column, I can see that
2 and 3 are the differences of the corresponding elements in Columns 2 and 3. But what’s happening in the third row? Did you mean for the third element to be 4, so that the first element in the row would also be the difference of the other two columns?

8

Only moderators can edit. For a list of other functions that have been disabled for mere mortals, see this sticky thread in ATMB: http://boards.straightdope.com/sdmb/showthread.php?s=&threadid=103503

9

I think ultrafilter’s point was that you can only combine rows with other rows and columns with other columns. In the example, the first column is a combination of the three rows and the determinant was 4, not 0. It kind of looks like you’ve been assuming that all along, but it’s good to be thorough.

10

Thanks much. RGillen explained what I meant better than I would, so we’ll go with that explanation.