MHB Can you solve this system of equations with four variables?

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The discussion centers around solving a system of four equations involving variables a, b, c, and d. The equations include a linear equation, a quadratic equation, a cubic equation, and a product equation. Participants explore various methods to find real solutions, with one proposed solution referenced from an online source. The complexity of the equations suggests that traditional algebraic methods may be insufficient, prompting discussions on numerical or graphical approaches. Ultimately, the thread seeks to identify all possible real solutions to the given system.
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Find all real solutions to the following system of equations:

$a+b+c+d=5$

$ab+bc+cd+da=4$

$abc+bcd+cda+dab=3$

$abcd=-1$
 
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Solution that I saw somewhere online:

We're given

$a+b+c+d=5$

$ab+bc+cd+da=4$

$abc+bcd+cda+dab=3$

$abcd=-1$

Let $X=a+c$ and $Y=b+d$. Then the system of equations is equivalent to

$x+Y=5$

$XY=4$

$Xbd+Yac=3$

$(Xbd)(Yac)=-4$

The first two of these equations imply $(X,\,Y)=(1,\,4)$ and the last two give $(Xbd,\,Yac)=(4,\,-1)$.

And this yields:

[TABLE="class: grid, width: 500"]
[TR]
[TD]$X$[/TD]
[TD]$Y$[/TD]
[TD]$Xbd$[/TD]
[TD]$Yac$[/TD]
[TD]$(a,\,c)$[/TD]
[TD]$(b,\,d)$[/TD]
[/TR]
[TR]
[TD]1[/TD]
[TD]4[/TD]
[TD]4[/TD]
[TD]-1[/TD]
[TD]$\dfrac{1\pm 2}{2}$[/TD]
[TD]2[/TD]
[/TR]
[TR]
[TD]1[/TD]
[TD]4[/TD]
[TD]-1[/TD]
[TD]4[/TD]
[TD]-[/TD]
[TD]-[/TD]
[/TR]
[TR]
[TD]4[/TD]
[TD]1[/TD]
[TD]4[/TD]
[TD]-1[/TD]
[TD]-[/TD]
[TD]-[/TD]
[/TR]
[TR]
[TD]4[/TD]
[TD]1[/TD]
[TD]-1[/TD]
[TD]4[/TD]
[TD]2[/TD]
[TD]$\dfrac{1\pm 2}{2}$[/TD]
[/TR]
[/TABLE]
 
Thread 'Erroneously finding discrepancy in transpose rule'

Thread 'Erroneously finding discrepancy in transpose rule'

Obviously, there is something elementary I am missing here. To form the transpose of a matrix, one exchanges rows and columns, so the transpose of a scalar, considered as (or isomorphic to) a one-entry matrix, should stay the same, including if the scalar is a complex number. On the other hand, in the isomorphism between the complex plane and the real plane, a complex number a+bi corresponds to a matrix in the real plane; taking the transpose we get which then corresponds to a-bi...
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