Are these right? (Determinants)

• February 26th 2010, 10:21 AM
StonerPenguin
Are these right? (Determinants)
$\left|\begin{array}{ccc}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{array}\right|$
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! (Rofl) So much fun :D

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

Evaluate the following determinants.
$\left|\begin{array}{ccc}-5 & 0 & 0 \\ 0 & 3 & -2 \\ 0 & 4 & 0 \end{array}\right|$
and
$3\left|\begin{array}{ccc}-2 & -1 & 0 \\ -3 & 5 & -2 \\ 0 & 8 & -1 \end{array}\right|$

$\left|\begin{array}{ccc}5 & 0 & 0 \\ 0 & 3 & -2 \\ 0 & 4 & 0 \end{array}\right|$
$=(5)(3)+(4)+(-2)-(3)-(-5)(4)(-2)=-56$
and
$3\left|\begin{array}{ccc}-2 & -1 & 0 \\ -3 & 5 & -2 \\ 0 & 8 & -1 \end{array}\right|$
$=3(-2\left|\begin{array}{cc}5 & -2 \\ 8 & -1 \end{array}\right|$ $+1\left|\begin{array}{cc}-3 & -2 \\ 0 & -1 \end{array}\right|$ $+0\left|\begin{array}{cc}-3 & 5 \\ 0 & 8 \end{array}\right|)$
$=3(-22 + 3 + 0) = -57$

And one more question;
Expand by minors
$\left|\begin{array}{ccc}3 & 4 & -2 \\ 6 & 4 & 3 \\ 0 & 4 & 2 \end{array}\right|$
This question really bothers 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;
$=3\left|\begin{array}{cc}4 & 3 \\ 4 & 2 \end{array}\right|$ $-4\left|\begin{array}{cc}6 & 3 \\ 0 & 2 \end{array}\right|$ $-2\left|\begin{array}{cc}6 & 4 \\ 0 & 4 \end{array}\right|$
$=3(8-12)-4(12-0)-2(24-0)=-108$

And if I expand the middle column;
Example, expanding the first row;
$=-4\left|\begin{array}{cc}6 & 3 \\ 0 & 2 \end{array}\right|$ $+4\left|\begin{array}{cc}3 & -2 \\ 0 & 2 \end{array}\right|$ $-4\left|\begin{array}{cc}3 & -2 \\ 6 & 3 \end{array}\right|$
$=-4(12-0)+4(6-0)-4(9-12)=-12$

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

Thanks in advance for any help
• February 26th 2010, 10:37 AM
Rapha
Hi

Quote:

Originally Posted by StonerPenguin
$\left|\begin{array}{ccc}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{array}\right|$
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! (Rofl) So much fun :D

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

Evaluate the following determinants.
$\left|\begin{array}{ccc}5 & 0 & 0 \\ 0 & 3 & -2 \\ 0 & 4 & 0 \end{array}\right|$
and
$3\left|\begin{array}{ccc}-2 & -1 & 0 \\ -3 & 5 & -2 \\ 0 & 8 & -1 \end{array}\right|$

$\left|\begin{array}{ccc}5 & 0 & 0 \\ 0 & 3 & -2 \\ 0 & 4 & 0 \end{array}\right|$
$=(5)(3)+(4)+(-2)-(3)-(-5)(4)(-2)=-56$

No, how did you calculate it anyway?

Have you ever heard of "rule of Sarrus". I get : det() = +40

Quote:

Originally Posted by StonerPenguin
and

$3\left|\begin{array}{ccc}-2 & -1 & 0 \\ -3 & 5 & -2 \\ 0 & 8 & -1 \end{array}\right|$
$=3(-2\left|\begin{array}{cc}5 & -2 \\ 8 & -1 \end{array}\right|$ $+1\left|\begin{array}{cc}-3 & -2 \\ 0 & -1 \end{array}\right|$ $+0\left|\begin{array}{cc}-3 & 5 \\ 0 & 8 \end{array}\right|)$
$=3(-22 + 3 + 0) = -57$

This is correct, well done

Quote:

Originally Posted by StonerPenguin
And one more question;
Expand by minors
$\left|\begin{array}{ccc}3 & 4 & -2 \\ 6 & 4 & 3 \\ 0 & 4 & 2 \end{array}\right|$
This question really bothers me since supposedly it doesn't matter with row or column you expan you're supposed to get the same answer everytime and I get a different answer everytime.

Example, expanding the first row;
$=3\left|\begin{array}{cc}4 & 3 \\ 4 & 2 \end{array}\right|$ $-4\left|\begin{array}{cc}6 & 3 \\ 0 & 2 \end{array}\right|$ $-2\left|\begin{array}{cc}6 & 4 \\ 0 & 4 \end{array}\right|$
$=3(8-12)-4(12-0)-2(24-0)=-108$

This is correct, too

Quote:

Originally Posted by StonerPenguin
And if I expand the middle column;
Example, expanding the first row;
$=-4\left|\begin{array}{cc}6 & 3 \\ 0 & 2 \end{array}\right|$ $+4\left|\begin{array}{cc}3 & -2 \\ 0 & 2 \end{array}\right|$ $-4\left|\begin{array}{cc}3 & -2 \\ 6 & 3 \end{array}\right|$
$=-4(12-0)+4(6-0)-4(9-12)$

No, it is $-4\left|\begin{array}{cc}3 & -2 \\ 6 & 3 \end{array}\right| = -4(9-(-12)) = -4(9+12)$

So it should be -4*21+24-48 = -108

Quote:

Originally Posted by StonerPenguin
So... what am I doing wrong? D:

Thanks in advance for any help

By the way, nice solution!

Rapha
• February 26th 2010, 12:28 PM
StonerPenguin
Ah thank you so much! Yeah, I had a serious case of brainfart here, how embarassing O: For the first question I used $a_1 b_2 c_3 + a_2 b_3 c_1 + a_3 b_1 c_2 - a_3 b_2 c_1 - a_2 b_1 c_3 - a_1 b_3 c_2$
but like an idiot I cast out some of the zeros D: e.g. $(5)(3)(0)=0$ not 15. However re-evaluating it, I get NEG 40, because
$(5)(3)(0) + (0)(4)(0) + (0)(0)(-2) - (0)(3)(0) - (0)(0)(0) - (-5)(4)(-2)$

So then det = $- (-5)(4)(-2)= -40$ Right? Or am I being stupid again? (Nerd)
• February 26th 2010, 12:42 PM
running-gag
Quote:

Originally Posted by StonerPenguin
Ah thank you so much! Yeah, I had a serious case of brainfart here, how embarassing O: For the first question I used $a_1 b_2 c_3 + a_2 b_3 c_1 + a_3 b_1 c_2 - a_3 b_2 c_1 - a_2 b_1 c_3 - a_1 b_3 c_2$
but like an idiot I cast out some of the zeros D: e.g. $(5)(3)(0)=0$ not 15. However re-evaluating it, I get NEG 40, because
$(5)(3)(0) + (0)(4)(0) + (0)(0)(-2) - (0)(3)(0) - (0)(0)(0) - (-5)(4)(-2)$

So then det = $- (-5)(4)(-2)= -40$ Right? Or am I being stupid again? (Nerd)

It is actually
$(5)(3)(0) + (0)(4)(0) + (0)(0)(-2) - (0)(3)(0) - (0)(0)(0) - (5)(4)(-2)$
because $a_1=5$
• February 26th 2010, 12:52 PM
StonerPenguin
Quote:

Originally Posted by running-gag
It is actually
$(5)(3)(0) + (0)(4)(0) + (0)(0)(-2) - (0)(3)(0) - (0)(0)(0) - (5)(4)(-2)$
because $a_1=5$

Ah crap! I wrote my original question wrong! The question was $\left|\begin{array}{ccc}-5 & 0 & 0 \\ 0 & 3 & -2 \\ 0 & 4 & 0 \end{array}\right|$ so $a_1 = -5$ I'm so sorry you guys, I'm usually not this bad about making mistakes D: Like I said, I think I scrambled my brains a bit, I stayed up 22 hours straight after only 4 hours of sleep (Nice excuse amirite? I'm retarded BAAWWW)

So it is -40 then right? Thank you so much for helping my stupid ass D':
• February 26th 2010, 12:59 PM
running-gag
Yes -40 is the right result (Wink)
• February 26th 2010, 03:04 PM
satx
Looks like you need to lay off that reefer, son ^_^