HOw can this determinant be 0 ?

[zetafunction]

HOw can this determinant be 0 ??

let be a 3x3 matrix with elements

$$a_{1,j}= j$$

$$a_{2,j}= j+3$$

$$a_{3,j}= 6+j$$

with i=1,2,3 and j=1,2,3 ...

but DetA=0 apparently there is no zeros or other condition that points that determinant should be 0, is there any explanation ??

 

[Moo Of Doom]

Because the rows aren't linearly independent:

2*[4 5 6] - 1*[1 2 3] = [7 8 9]

 

[zgozvrm]

The determinant of a 3x3 matrix

a b c
d e f
g h i

is: aei+bfg+cdh-afh-bdi-ceg


In your case

a b c
d e f =
g h i

1 2 3
4 5 6
7 8 9



Therefore the determinant is:
1x5x9 + 2x6x7 + 3x4x8 - 1x6x8 - 2x4x9 - 3x5x7
= 45 + 84 + 96 - 48 -72 - 105
= 225 - 225
= 0

 

[Gear300]

If you look at the 3x3 matrix as an array of 3 vectors (row or column), all of them are parallel.

 

[g_edgar]

If you look at the 3x3 matrix as an array of 3 vectors (row or column), all of them are parallel.

?? [1,2,3] and [4,5,6] are parallel?

 

[HallsofIvy]

If you look at the 3x3 matrix as an array of 3 vectors (row or column), all of them are parallel.

No, they aren't. But they all lie in the same plane. Another way of looking at this is that that the determinant of a matrix having three vectors, u, v, w, say, as rows (or columns) is a "triple product": $u\cdot(v\times w)$. And that can be interpreted as the volume of the parallelopiped having the three vectors as edges. That volume is 0 if and only if the three vectors are all in the same plane so that the parallelopiped has no "height".

 

[Gear300]

?? [1,2,3] and [4,5,6] are parallel?

No, they aren't. But they all lie in the same plane. Another way of looking at this that that the determinant of a matrix having three vectors, u, v, w, say, as rows (or columns) is a "triple product": $u\cdot(v\times w)$. And that can be interpreted as the volume of the parallelopiped having the three vectors as edges. That volume is 0 if and only if the three vectors are all in the same plane so that the parallelopiped has no "height".

Sorry about that...mixed up the words...they lie on the same plane as HallsofIvy stated.