If we have a system:
$$
\left\{
\begin{array}{l}
ax+by+z = 1\\
x+aby+z=b\\
x+by+az=1
\end{array}
\right.
$$
What would be the best way to discuss it?
Here's how I started (I used Kronecker–Capelli theorem):
First we create a cofactor matrix:
$$(A|B) = \left(\begin{array}{ccc|c}
a & b & 1 & 1\\
1 & ab & 1 & b\\
1 & b & a & 1
\end{array}
\right)
$$
Then we try to get rank of the matrix to check the system consistency in different cases:
$$rank(A|B) = rank\left(\begin{array}{ccc|c}
a & b & 1 & 1\\
1 & ab & 1 & b\\
1 & b & a & 1
\end{array}
\right)=rank
\left(\begin{array}{ccc|c}
a & b & 1 & 1\\
0 & b(a-1)(a+1) & a-1 & ab-1\\
0 & b(a-1) & (a-1)(a+1) & a-1
\end{array}
\right)
$$
Now we can see that for $a=1$ we have to cases:
1) if $b≠1$, the system is inconsistent.
2) if $b=1$ we have a two-parametric solution:
$$x+y+z=1$$ $$x=1-y-z\\y,z∈R$$
so
$$(x,y,z)=(1-y-z, y, z); y,z∈R$$
Now, what happens if $a≠1$? I got stuck here and I'm not sure where the condition will lead me? Is there maybe a better way of discussing this system?
And now, the biggest question: What do each of the cases mean geometrically? If the system is inconsistent, does it mean that the planes are parallel? (lol)
What does it mean if the system has infinite number of solutions and what does it mean if there's only one solution? Thank you for your time.
Best Answer
Geometrically, it may mean that one of the planes is parallel to one of the others, or perhaps none of them are parallel, but they form an infinite triangular prism (i.e. each pair of planes intersects in a line, producing three parallel lines).
In terms of what to do in the case of $a \neq 1$, I would consider the case $a = 0$, then I would consider the case where $a$ is not equal to $0$ or $1$. In this case, you are allowed to divide by $a - 1$ and $a$, so I would divide the top row by $a$, and the middle and bottom rows by $a - 1$.