Question

Sigma Notation Question: Is there any way to put this summation into Sigma Notation? I'm trying to make this a bit more succinct because I will eventually have to nest this to have 27 terms.
\[\begin{vmatrix}a_{11} & 0 & 0 \\a_{21} & a_{22} & a_{23}\\a_{31} & a_{32} & a_{33} \end{vmatrix} + \begin{vmatrix}0 & a_{12} & 0 \\a_{21} & a_{22} & a_{23}\\a_{31} & a_{32} & a_{33} \end{vmatrix} + \begin{vmatrix}0 & 0 & a_{13} \\a_{21} & a_{22} & a_{23}\\a_{31} & a_{32} & a_{33} \end{vmatrix}\]

Answer 1

so your gonna change only the 1st row like 1st u have a_{11} then a_{12} and u keep shifting them and u want summation till a_{27} right?

Answer 2

Basically, that's the first level of sums. So, take the first matrix there. I need to split that up even further like so:
\[\begin{vmatrix}a_{11} & 0 & 0 \\a_{21} & a_{22} & a_{23}\\a_{31} & a_{32} & a_{33} \end{vmatrix} = \begin{vmatrix}a_{11} & 0 & 0 \\a_{21} & 0 & 0\\a_{31} & a_{32} & a_{33} \end{vmatrix} + \begin{vmatrix}a_{11} & 0 & 0 \\0 & a_{22} & 0\\a_{31} & a_{32} & a_{33} \end{vmatrix} + \begin{vmatrix}a_{11} & 0 & 0 \\0 & 0 & a_{23}\\a_{31} & a_{32} & a_{33} \end{vmatrix}\]After I do that for the first 2 (I have 9 matrices at this point), for each of those 9, I need to do it again on the third row to get 27 matrices, so you can see why I want to try to use Sigma notation.

Answer 3

ok the summation is 0

Answer 4

note that evry determinant will have 2 zeroes in 2 diff columns

whenever u see two 0s in two different columns then the determinant equal 0

can u find out y is it so :)

Answer 5

I'm not trying to calculate it though. I'm trying to write the proof of evaluating a 3x3 determinant using this method for class. Also, it wouldn't be 0 because there would still be one entry in every row, wouldn't there, which can row reduce to an upper triangular matrix?

Answer 6

Do you think it'd be fine to denote it as row vectors like this:

Let \(v_1 = \begin{pmatrix}a_{11} & a_{12} & a_{13}\end{pmatrix}\), \(v_2 = \begin{pmatrix}a_{21} & a_{22} & a_{23}\end{pmatrix}\), \(v_3 = \begin{pmatrix}a_{31} & a_{32} & a_{33}\end{pmatrix}\)
\[\begin{vmatrix}a_{11} & a_{12} & a_{13} \\a_{21} & a_{22} & a_{23}\\a_{31} & a_{32} & a_{33} \end{vmatrix} = \begin{vmatrix}v_1 \\ v_2 \\ v_3\end{vmatrix}\]And letting \(t_i\) represent the ith permutation matrix row, \[\begin{vmatrix}a_{11} & 0 & 0 \\a_{21} & a_{22} & a_{23}\\a_{31} & a_{32} & a_{33} \end{vmatrix} = \begin{vmatrix}t_1 \\ v_2 \\ v_3\end{vmatrix}\]Would this work to make it into a sigma, or is there a cleaner way?