3x3 matrix with complex numbers

[CJovaras]

Homework Statement

Solve for x and y if

| 1 1+2j 0 |
| 1-j -1 -1 | = 2j
| x 0 yj |


j represents imaginary numbers and x and y represent variables. My entire class is struggling with understanding this.

Relevant Equations

none

The attempt at a solution:

I tried the normal method to find the determinant equal to 2j. I ended up with:
2j = -4yj -2xj -2j -x +y

then I tried to see if I had to factorize with j so I didn't turn the j^2 into -1 and ended up with 2 different options:

1) 0= y(-4j-j^2) -x(2j-1) -2j

2) -2j(2y +x +1) +y -x

After that I get stuck, I've spent at least 8 hours staring at this problem and googling how to find a solution but nothing makes sense to me.

 

[fresh_42]

Homework Statement:: Solve for x and y if

| 1 1+2j 0 |
| 1-j -1 -1 | = 2j
| x 0 yj |


j represents imaginary numbers and x and y represent variables. My entire class is struggling with understanding this.
Relevant Equations:: none

The attempt at a solution:

I tried the normal method to find the determinant equal to 2j. I ended up with:
2j = -4yj -2xj -2j -x +y

Let's see.
$$
\begin{align*}
\det\begin{pmatrix}1&1+2i&0\\1-i&-1&-1\\x&0&yi\end{pmatrix}&=-iy-(1+2i)((1-i)iy+x)\\
&=-iy-(1+2i)(iy+y+x)\\&=-iy-iy-y-x+2y-2iy-2ix\\&=(-x-y)+i(-2x-4y)
\end{align*}
$$
Looks as if one of us made a mistake. (I developed the determinant along the first row.)

Edit (for future readers): Correction (see discussion below):
the determinant is $(-x+y)+i(-2x-4y)$ since I have forgotten to add the $+2y$ term.



Here is explained how you can type formulas on PF: https://www.physicsforums.com/help/latexhelp/

then I tried to see if I had to factorize with j so I didn't turn the j^2 into -1 and ended up with 2 different options:

1) 0= y(-4j-j^2) -x(2j-1) -2j

2) -2j(2y +x +1) +y -x

After that I get stuck, I've spent at least 8 hours staring at this problem and googling how to find a solution but nothing makes sense to me.

 

[DrClaude]

Let's see.
$$
\begin{align*}
\det\begin{pmatrix}1&1+2i&0\\1-i&-1&-1\\x&0&yi\end{pmatrix}&=-iy-(1+2i)((1-i)iy+x)\\
&=-iy-(1+2i)(iy+y+x)\\&=-iy-iy-y-x+2y-2iy-2ix\\&=(-x-y)+i(-2x-4y)
\end{align*}
$$
Looks as if one of us made a mistake. (I developed the determinant along the first row.)

You made a mistake too on the sign of the first y
$$
\ldots =
(-x+y)+i(-2x-4y)
$$

 

[berkeman]

Here is explained how you can type formulas on PF: https://www.physicsforums.com/help/latexhelp/

@CJovaras -- you can also see the LaTeX that fresh_42 used for his math. Right-click your mouse over the LaTeX and "copy" it to your clipboard, then paste it into a *.txt file to study.

 

[DrClaude]

2) -2j(2y +x +1) +y -x

Assuming you get the correct equation here, you can then continue by comparing real and imaginary parts. For instance, you will get $y-x = 0$ by comparing the real parts. You can then use that to solve for $x$ using the imaginary parts.

 

[fresh_42]

You made a mistake too on the sign of the first y
$$
\ldots =
(-x+y)+i(-2x-4y)
$$

I did not. It is $(-1) \cdot (-i) \cdot (iy) = (-1)^2 i^2 y = -y$

 

[DrClaude]

I did not.

$$
\begin{align*}
\det\begin{pmatrix}1&1+2i&0\\1-i&-1&-1\\x&0&yi\end{pmatrix}&=-iy-(1+2i)((1-i)iy+x)\\
&=-iy-(1+2i)(iy+y+x)\\&=-iy-iy-y-x+2y-2iy-2ix\\&=(-x-y)+i(-2x-4y)
\end{align*}
$$

You have $-y$ and $+2y$ on the penultimate row:
$$
-iy-iy-y-x+2y-2iy-2ix = (-x+y)+i(-2x-4y)
$$

 

[DrClaude]

And you ignored the minus in front of $(1+2i)$!

I don't want us to derail this thread, but
(1) the last two lines in post #2 do not form an equality, so one of them is incorrect

(2) when I calculate the determinant (independent of what you wrote) I get (-x+y)+i(-2x-4y)

 

[fresh_42]

I don't want us to derail this thread, but
(1) the last two lines in post #2 do not form an equality, so one of them is incorrect

(2) when I calculate the determinant (independent of what you wrote) I get (-x+y)+i(-2x-4y)

I didn't make a sign error! I lost a complete term ($2y$)! That's why I didn't understand you.

 

[CJovaras]

@CJovaras -- you can also see the LaTeX that fresh_42 used for his math. Right-click your mouse over the LaTeX and "copy" it to your clipboard, then paste it into a *.txt file to study.

thank you!!!

 

[Mark44]

Homework Statement:: Solve for x and y if

| 1 1+2j 0 |
| 1-j -1 -1 | = 2j
| x 0 yj |


j represents imaginary numbers and x and y represent variables. My entire class is struggling with understanding this.
Relevant Equations:: none

The attempt at a solution:

I tried the normal method to find the determinant equal to 2j. I ended up with:
2j = -4yj -2xj -2j -x +y

You're very close here, but you have an extra term. The equation from the determinant simplifies to -4yj -2xj -x +y = 2j. Your equation has an extra term of -2j in it. Calculating the determinant is very tedious and you can easily make a mistake.
Check your determinant again.

then I tried to see if I had to factorize with j so I didn't turn the j^2 into -1 and ended up with 2 different options:

It's a mistake to not replace $j^2$ by -1.

1) 0= y(-4j-j^2) -x(2j-1) -2j

2) -2j(2y +x +1) +y -x

After that I get stuck, I've spent at least 8 hours staring at this problem and googling how to find a solution but nothing makes sense to me.

As it turns out, x and y have the same value, but I won't say more than that. Once I got my solution, I replaced x and y with the values I found, and calculated the determinant. My result was 2j, which verified that my solution was correct. As mentioned, calculating the determinant was tedious, so I wrote a Python program to do the grunt work for me, using the SymPy library.

 

[CJovaras]

You're very close here, but you have an extra term. The equation from the determinant simplifies to -4yj -2xj -x +y = 2j. Your equation has an extra term of -2j in it. Calculating the determinant is very tedious and you can easily make a mistake.
Check your determinant again.It's a mistake to not replace $j^2$ by -1.

As it turns out, x and y have the same value, but I won't say more than that. Once I got my solution, I replaced x and y with the values I found, and calculated the determinant. My result was 2j, which verified that my solution was correct. As mentioned, calculating the determinant was tedious, so I wrote a Python program to do the grunt work for me, using the SymPy library.

Please tell me you're joking?!?! I found an answer where x and y were equal to one another in the first 30 minutes of working on the problem but I had assumed I had made a mistake! Thank you lol.

 

[WWGD]

Wolfram can do this for you. Edit: I'm on my phone, hard to switch back-forth between sites.

 

[fresh_42]

Wolfram can do this for you. Edit: I'm on my phone, hard to switch back-forth between sites.

https://www.symbolab.com/solver/matrix-calculator is better. But I think it is important to know the formula instead of using websites. They should be used after people learned all about determinants, their computation included.

 

[WWGD]

Now, a more interesting question to me is the interpretation of a Complex determinant. Can't use volume, area, etc. And linear dependence in Complexes is also different in a Complex vector space ( Complexes over themselves).

 

[fresh_42]

Now, a more interesting question to me is the interpretation of a Complex determinant. Can't use volume, area, etc. And linear dependence in Complexes is also different in a Complex vector space ( Complexes over themselves).

My thoughts about this are here:
https://www.physicsforums.com/insights/an-overview-of-complex-differentiation-and-integration/

The apparent ambiguity occurs only because we insist that volumes are real. Let them be complex and everything is fine. We have complex Graßmann algebras, too.

If we want real volumes, then forget about complex integration.

 

[WWGD]

Volumes are Real, but their names have been changed to protect the innocent;).

 

[WWGD]

How about an argument using Complexification, or Extension by Scalars?

 

[fresh_42]

How about an argument using Complexification, or Extension by Scalars?

I think this discussion now exceeds the limits of this thread tremendously.

Time to close the debate here.