- #1
Brutus
- 7
- 0
Why am I not allowed to reuse rows to find the determinant via elementary operations?
Hi,
I am learning about matrices and determinants and there is something I can't figure out, straight to the point with an example:
Evaluating the determinant...
[tex]
\begin{bmatrix}
1&2&3&4 \\
5&6&7&8 \\
2&6&4&8 \\
3&1&1&2
\end{bmatrix}=2
\begin{bmatrix}
1&2&3&4 \\
5&6&7&8 \\
1&3&2&4 \\
3&1&1&2
\end{bmatrix}=2
\begin{bmatrix}
1&2&3&4 \\
0&-4&-8&-12 \\
0&1&-1&0 \\
0&-5&-8&-10
\end{bmatrix}=8
\begin{bmatrix}
1&2&3&4 \\
0&-1&-2&-3 \\
0&1&-1&0 \\
0&-5&-8&-10
\end{bmatrix}=8
\begin{bmatrix}
1&2&3&4 \\
0&-1&-2&-3 \\
0&0&-\frac{1}{2}&-2 \\
0&0&-\frac{1}{2}&0
\end{bmatrix}=8
\begin{bmatrix}
1&2&3&4 \\
0&-1&-2&-3 \\
0&0&-\frac{1}{2}&-2 \\
0&0&0&2
\end{bmatrix}=8
[/tex]
Obviously, it's wrong, the right answer is 72(see [1]), I didn't work on the main diagonal on purpose, I wanted to see if I was allowed to reuse rows(or cols), so why am I not allowed? what am I really doing to the matrix each time I reuse a row(or col) ?
I am not asking for a proof or anything, just a simple explanation for human beings ;).
Also, working with both rows and columns is not allowed either, I guess this is a particular case of the above since I am reusing a cell.
Thank you.
[1] http://www.sosmath.com/matrix/determ1/determ1.html
Hi,
I am learning about matrices and determinants and there is something I can't figure out, straight to the point with an example:
Evaluating the determinant...
[tex]
\begin{bmatrix}
1&2&3&4 \\
5&6&7&8 \\
2&6&4&8 \\
3&1&1&2
\end{bmatrix}=2
\begin{bmatrix}
1&2&3&4 \\
5&6&7&8 \\
1&3&2&4 \\
3&1&1&2
\end{bmatrix}=2
\begin{bmatrix}
1&2&3&4 \\
0&-4&-8&-12 \\
0&1&-1&0 \\
0&-5&-8&-10
\end{bmatrix}=8
\begin{bmatrix}
1&2&3&4 \\
0&-1&-2&-3 \\
0&1&-1&0 \\
0&-5&-8&-10
\end{bmatrix}=8
\begin{bmatrix}
1&2&3&4 \\
0&-1&-2&-3 \\
0&0&-\frac{1}{2}&-2 \\
0&0&-\frac{1}{2}&0
\end{bmatrix}=8
\begin{bmatrix}
1&2&3&4 \\
0&-1&-2&-3 \\
0&0&-\frac{1}{2}&-2 \\
0&0&0&2
\end{bmatrix}=8
[/tex]
Obviously, it's wrong, the right answer is 72(see [1]), I didn't work on the main diagonal on purpose, I wanted to see if I was allowed to reuse rows(or cols), so why am I not allowed? what am I really doing to the matrix each time I reuse a row(or col) ?
I am not asking for a proof or anything, just a simple explanation for human beings ;).
Also, working with both rows and columns is not allowed either, I guess this is a particular case of the above since I am reusing a cell.
Thank you.
[1] http://www.sosmath.com/matrix/determ1/determ1.html
Last edited: