# Why am I not allowed to reuse rows when operating matrices via elementary operations?

1. Sep 22, 2008

### Brutus

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...
$$\begin{bmatrix} 1&2&3&4 \\ 5&6&7&8 \\ 2&6&4&8 \\ 3&1&1&2 \end{bmatrix}=2 \begin{bmatrix} 1&2&3&4 \\ 5&6&7&8 \\ 1&3&2&4 \\ 3&1&1&2 \end{bmatrix}=2 \begin{bmatrix} 1&2&3&4 \\ 0&-4&-8&-12 \\ 0&1&-1&0 \\ 0&-5&-8&-10 \end{bmatrix}=8 \begin{bmatrix} 1&2&3&4 \\ 0&-1&-2&-3 \\ 0&1&-1&0 \\ 0&-5&-8&-10 \end{bmatrix}=8 \begin{bmatrix} 1&2&3&4 \\ 0&-1&-2&-3 \\ 0&0&-\frac{1}{2}&-2 \\ 0&0&-\frac{1}{2}&0 \end{bmatrix}=8 \begin{bmatrix} 1&2&3&4 \\ 0&-1&-2&-3 \\ 0&0&-\frac{1}{2}&-2 \\ 0&0&0&2 \end{bmatrix}=8$$

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: Sep 22, 2008
2. Sep 22, 2008

### Dick

Re: Why am I not allowed to reuse rows when operating matrices via elementary operati

You can reuse rows. You can also mix row and column operations. But how did you get from the fourth matrix to the fifth? I'm baffled.

3. Sep 23, 2008

### Brutus

Re: Why am I not allowed to reuse rows when operating matrices via elementary operati

Nevermind, I made a stupid mistake, in step 4 I did: row 3 - row 1 * 1/2 and forgot to write the -1/2 in A<sub>3 1</sub> ...
I also forgot to include the 5/2 in A<sub>4 1</sub> when doing row 4 - row 1 * (-5/2)
damn...

I guess my mind was expecting a failure and it tricked itself into it...

Thank you.

Last edited: Sep 23, 2008
4. Sep 23, 2008

### HallsofIvy

Staff Emeritus
Re: Why am I not allowed to reuse rows when operating matrices via elementary operati

You can, as Dick said, use a row more than once. But it is more orderly and less error-prone if you work from upper left to lower right working with each row and column in turn. I would have done this as follows:
$$\begin{bmatrix} 1&2&3&4 \\5&6&7&8 \\2&6&4&8 \\3&1&1&2\end{bmatrix}= \begin{bmatrix}1&2&3&4\\0&-4&-8&-12\\0&2&-2&0\\0&-5&-8&-10\end{bmatrix}$$
"clearing" the first column. Now, seeing the "-4" in the pivot for the second column/row,
$$=-4\begin{bmatrix}1&2&3&4\\0&1&2&3\\0&0&-6&-6\\0&0&2&5\end{bmatrix}$$
$$=(-4)(-6)\begin{bmatrix}1&2&3&4\\0&1&2&3\\0&0&1&1\\0&0&0&3\end{bmatrix}$$
$$=(-4)(-6)(3)\begin{bmatrix}1&2&3&4\\0&1&2&3\\0&0&1&1\\0&0&0&1\end{bmatrix}$$
and now, since we have "1"s along the main diagonal, the determinant is (-4)(-6)(3)= 72.

5. Sep 23, 2008

### Brutus

Re: Why am I not allowed to reuse rows when operating matrices via elementary operati

Great, thanks.