Why does the determinant always equal zero for a matrix of consecutive numbers?

#1
Hi. Why does the determinant always equal zero for a matrix of consecutive numbers?

This applies whether the consecutive numbers are in the matrix starting from smallest to largest, or vice versa. It also applies irrespective of whether they are entered row then column or vice versa, which makes sense, I guess.

eg

123
456
789

and

987
654
321

and

147
258
369

all equal 0

If the same group of consecutive numbers are scattered around the matrix however, the determinant will not equal zero.

Oh, and while I have only tested this on a 3x3 matrix, I assume it holds true for any square matrix.

Don't lose any sleep over this one, as I am just asking out of curiousity while trying to get my head around matrix algebra.

Also, does anyone have a GOOD definition of what a determinant actually is?

I understand how it is used in equations, but not what it actually IS.

Thank you.
 
Last edited:

BGM

TS Contributor
#2
http://en.wikipedia.org/wiki/Determinant#Properties_characterizing_the_determinant

Note the property 4: The determinant is unchanged if you adding a multiple of one row to another row, or a multiple of one column to another column.

So in general if you have an arithmetic sequence, e.g. in a 3 by 3 matrix, then

\( \left|\begin{matrix} a + d && a + 2d && a + 3d \\
a + 4d && a + 5d && a + 6d \\
a + 7d && a + 8d && a + 9d \end{matrix}\right|
= \left|\begin{matrix} a + d && a + 2d && a + 3d \\
a + 4d && a + 5d && a + 6d \\
3d && 3d && 3d \end{matrix}\right| \)

\( = \left|\begin{matrix} a + d && a + 2d && a + 3d \\
3d && 3d && 3d \\
3d && 3d && 3d \end{matrix}\right|
= \left|\begin{matrix} a + d && a + 2d && a + 3d \\
3d && 3d && 3d \\
0 && 0 && 0 \end{matrix}\right| \)

i.e. You can always obtain a zero row/column in the process, which shows that the matrix has zero determinant/no full rank.
 
#3
Thanks BGM.
Thanks, that makes (some sort) of sense to me.
It'll just take me some time to digest it. :)
Thanks again, I appreciate it.