$\displaystyle \left|\begin{array}{ccc}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{array}\right|$

Hello all! I'd like to check these as I'm not very confident that I got them right, and I'd really appreciate your input, especially as I've scrambled my brains with trilingual madness! So much fun

Anywho here's the questions with my answers, are they right?

Evaluate the following determinants.

$\displaystyle \left|\begin{array}{ccc}-5 & 0 & 0 \\ 0 & 3 & -2 \\ 0 & 4 & 0 \end{array}\right|$

and

$\displaystyle 3\left|\begin{array}{ccc}-2 & -1 & 0 \\ -3 & 5 & -2 \\ 0 & 8 & -1 \end{array}\right|$

My answers;

$\displaystyle \left|\begin{array}{ccc}5 & 0 & 0 \\ 0 & 3 & -2 \\ 0 & 4 & 0 \end{array}\right|$

$\displaystyle =(5)(3)+(4)+(-2)-(3)-(-5)(4)(-2)=-56$

and

$\displaystyle 3\left|\begin{array}{ccc}-2 & -1 & 0 \\ -3 & 5 & -2 \\ 0 & 8 & -1 \end{array}\right|$

$\displaystyle =3(-2\left|\begin{array}{cc}5 & -2 \\ 8 & -1 \end{array}\right|$$\displaystyle +1\left|\begin{array}{cc}-3 & -2 \\ 0 & -1 \end{array}\right|$$\displaystyle +0\left|\begin{array}{cc}-3 & 5 \\ 0 & 8 \end{array}\right|)$

$\displaystyle =3(-22 + 3 + 0) = -57$

And one more question;

Expand by minors

$\displaystyle \left|\begin{array}{ccc}3 & 4 & -2 \\ 6 & 4 & 3 \\ 0 & 4 & 2 \end{array}\right|$

This questionreallybothers me considering that supposedly it doesn't matter which row or column you expand- you're supposed to get the same answer everytime, I'm getting a different answer each time.

Example, expanding the first row;

$\displaystyle =3\left|\begin{array}{cc}4 & 3 \\ 4 & 2 \end{array}\right|$$\displaystyle -4\left|\begin{array}{cc}6 & 3 \\ 0 & 2 \end{array}\right|$$\displaystyle -2\left|\begin{array}{cc}6 & 4 \\ 0 & 4 \end{array}\right|$

$\displaystyle =3(8-12)-4(12-0)-2(24-0)=-108$

And if I expand the middle column;

Example, expanding the first row;

$\displaystyle =-4\left|\begin{array}{cc}6 & 3 \\ 0 & 2 \end{array}\right|$$\displaystyle +4\left|\begin{array}{cc}3 & -2 \\ 0 & 2 \end{array}\right|$$\displaystyle -4\left|\begin{array}{cc}3 & -2 \\ 6 & 3 \end{array}\right|$

$\displaystyle =-4(12-0)+4(6-0)-4(9-12)=-12$

So... what am I doing wrong? D:

Thanks in advance for any help