Here is the question: Factorise the determinant
$$\text{det}\begin{pmatrix}
z & 1 & 2 \\
1 & z & 3 \\
1 & 1 & z+1
\end{pmatrix}$$
and hence solve the system of equations
$$\left\{\begin{matrix}
zx+y=2\\
x+zy=3\\
x+y=z+1
\end{matrix}\right.$$
I've factored the determinant to $(z-1)(z^2+2z-4)$, but I fail to see how to use the determinant to solve this system.
How can the determinant actually be used for this? After doing some research into potential methods, I came across something called Cramer's Rule (although I do not believe that this would be the intended method). Not to mention, it seems to only be useable for linear/planar equations.
Best Answer
If you don't mind, I would like to offer a solution without using determinants. If we add the first two equations, $$x+y+zx+xy=5 \implies (x+y)(z+1)=5$$ From the third equation, $x+y=z+1$ $$\therefore (z+1)^2=5 \implies z=-1\pm\sqrt5 \implies x+y=-1\pm\sqrt5$$ You can solve for $x$ and $y$ using the above. Hope this helps.