Animated Solution for Mathematics - Matrices and Determinants: If the system of linear equations $$\begin{gathered} x+y+3z=0 \\ x+3y+k^{2}z=0 \\ 3x+y+3z=0 \end{gathered}$$ has a non-zero solution $(x,y,z)$ for some $k\in R$ then $x+(\frac{y}{z}{)}^2$ is equal to

We are presented with a system of three linear equations: $x+y+3z=0$ $x+3y+k^{2}z=0$ $3x+y+3z=0$

Notice that every equation is equal to zero. This is a homogeneous system. In the context of JEE mathematics, this is a critical signal.

A homogeneous system $Ax=0$ always possesses the trivial solution $(0,0,0)$. However, the problem specifies that the system has a non-zero solution. This implies that the system is linearly dependent, and the determinant of the coefficient matrix $\Delta$ must be equal to zero.

We construct the coefficient matrix and set its determinant to zero:

Expanding along the first row, we perform the calculation:

Simplifying the expression, we obtain:

This simplifies to $k^2=9$.

Often in JEE, the most elegant path avoids exhaustive calculation. Consider the first equation, $x+y+3z=0$, and the third equation, $3x+y+3z=0$.

Subtracting the first equation from the third causes the $y$ and $3z$ terms to vanish:

Substituting $x=0$ into the first equation yields $0+y+3z=0$, which simplifies to $y=-3z$. Therefore, the ratio is $\frac{y}{z}=-3$.

We now calculate the required expression $x+(\frac{y}{z}{)}^2$.

Substituting our derived values $x=0$ and $\frac{y}{z}=-3$:

The final answer is 9. Remember, the key to these problems is the ability to pause, observe the symmetry, and choose the path of least resistance.