Originally Posted by

**Mrs. White** I'm given his matrix and asked to find the determinant.

$\displaystyle \left[\begin{array}{ccccc}2&-2&0&0&-3\\3&0&3&2&-1\\0&1&-2&0&2\\-1&2&0&3&0\\0&4&1&0&0\end{array}\right]$

Now I do $\displaystyle -4C_{3} + C_{2} \longrightarrow C_{2}$which gives me this

$\displaystyle \left[\begin{array}{ccccc}2&-2&0&0&-3\\3&-12&3&2&-1\\0&9&-2&0&2\\-1&2&0&3&0\\0&0&1&0&0\end{array}\right]$

This eliminates the 5th row and 3rd column.

I'm now left with this:

$\displaystyle \left[\begin{array}{cccc}2&-2&0&-3\\3&-12&2&-1\\0&9&0&2\\-1&2&3&0\end{array}\right]$

Now I'm trying to get rid of row and column 1:

$\displaystyle R_{1} + R{2} \longrightarrow R_{2}$

$\displaystyle 3R_{1} + 2R{4} \longrightarrow R_{4}$ Awkward says: here you doubled the determinant.

$\displaystyle \left[\begin{array}{cccc}2&0&0&0\\3&-9&2&7\\0&9&0&4\\-1&1&3&-3\end{array}\right]$

And now I'm left with this:

$\displaystyle \left[\begin{array}{ccc}-9&2&7\\9&0&4\\1&3&-3\end{array}\right]$

Now I can solve this and this is what I get:

$\displaystyle -9 \left[\begin{array}{cc}0&4\\3&-3\end{array}\right]$ $\displaystyle -2 \left[\begin{array}{cc}9&4\\1&-3\end{array}\right]$ $\displaystyle +7 \left[\begin{array}{cc}9&0\\1&3\end{array}\right]$

That gives me:

$\displaystyle 2 [- 9(-12) - 2(-31) + 7(27)]= 2(359)$

This is where I don't understand because the answer in the book is 359, but what about the cofactor of 2 when I eliminated the row and column 1?

Don't I have to multiply everything by 2 which would not be the answer in the book?

Thank you.