Animated Solution for Mathematics - Matrices and Determinants: If the system of equations $2x+3y-z=0,x+ky-2z=0$ and $2x-y+z=0$ has a non-trivial solution $(x,y,z)$, then $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}+k$ is equal to :-

Welcome, future engineers. Today, we are not just solving a system of linear equations; we are peeling back the curtain on the elegant structure of linear algebra.

When you see a system like $2x+3y-z=0$, $x+ky-2z=0$, and $2x-y+z=0$, your first instinct might be to reach for Cramer's Rule or Gaussian elimination. But pause. Look at the right-hand side; it is all zeros.

This is a homogeneous system, and it behaves differently than anything else in the realm of linear equations.

The Determinant Gatekeeper

The problem demands a 'non-trivial solution.' In the world of homogeneous systems, the origin $(0,0,0)$ is always a solution—we call this the 'trivial' solution.

But we want more. We want the system to have a life of its own, independent of the origin. For this to happen, the matrix of coefficients $A$ must be singular.

Mathematically, this is our gatekeeper: the determinant $|A|$ must be zero. We set up our determinant:

Expanding this along the first row is a test of your focus. We calculate:

As we simplify, the terms dance into place: $2k-4-15+1+2k=0$. Suddenly, the complexity collapses into $4k-18=0$, revealing $k=\frac{9}{2}$. We have unlocked the first secret.

The Art of Ratios

Now that we have $k$, we need to evaluate the expression $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}+k$. Many students panic here, thinking they need to solve for $x,y,$ and $z$ individually.

But look closely at the expression. It is a ratio of variables. This is a huge hint! We don't need the absolute values; we only need the relative proportions.

We return to our equations. By adding the first and third equations, we witness a beautiful cancellation:

With $y$ in our pocket, we substitute it back into the third equation: $2x-(-2x)+z=0$, which yields $4x+z=0$, or $z=-4x$. We have expressed everything in terms of $x$. This is the power of linear dependence.

The Final Symphony

We are at the finish line. We have $y=-2x$, $z=-4x$, and $k=\frac{9}{2}$. Let us substitute these into our target expression:

Watch as the variable $x$ vanishes from every term, leaving us with pure numbers:

The $-\frac{1}{2}$ and $+\frac{1}{2}$ cancel out with poetic precision, leaving us with $-4+4.5$, which is exactly $\frac{1}{2}$.

You see? The complexity was just a mask. By understanding the underlying structure of the system, we turned a daunting algebraic problem into a simple, elegant calculation. Keep this mindset, and no problem will ever be too difficult for you.