Animated Solution for Mathematics - Matrices and Determinants: Let $a,\lambda ,\mu \in \mathbb{R}$. Consider the system of linear equations $ax+2y=\lambda ,3x-2y=\mu$. Which of the following statement(s) is (are) correct?

Welcome, fellow traveler on the path to JEE Advanced mastery. Today, we are not just solving a system of linear equations; we are peeling back the curtain on the geometry of linear algebra. We are given the system:

At first glance, this looks like a standard algebra problem. Imagine you are standing on a two-dimensional Cartesian plane where each equation represents a straight line. The variables $x$ and $y$ are the coordinates of points on these lines, while the coefficients $a,2,3,$ and $-2$ dictate their slopes and orientations.

The constants $\lambda$ and $\mu$ determine their positions relative to the origin. The fundamental question is: how do these two lines interact? Do they cross, run parallel forever, or are they actually the same line in disguise?

The Determinant

To understand the behavior of this system, we need a gatekeeper: the determinant of the coefficient matrix. Let us construct our matrix $A$ from the coefficients of $x$ and $y$:

The determinant, denoted by $\Delta$, is the mathematical "volume" or "scaling factor" of this transformation. It determines whether the system has a unique solution. We calculate it as:

The core principle is that if $\Delta eq 0$, the lines are not parallel. They have different slopes and must intersect at exactly one point, which is the "Unique Solution" territory. Setting $-2a - 6 eq 0$ leads us to the condition $a eq -3$.

As long as $a eq -3$, the system is perfectly well-behaved. This confirms that for any $a eq -3$, we have a unique solution, regardless of the values of $\lambda$ and $\mu$. This is the beauty of linear independence.

When $a=-3$

Now, let us step into the danger zone. What happens when $a=-3$? Our determinant $\Delta$ becomes zero, which geometrically means the lines have become parallel.

Parallel lines can behave in two very different ways: they can be distinct (never meeting, hence "no solution") or they can be coincident (lying on top of each other, hence "infinitely many solutions"). To distinguish between these, we must look at the constants $\lambda$ and $\mu$.

Substituting $a=-3$ back into our system yields:

The left-hand sides are exact negatives of each other. If we add these two equations together, the variables $x$ and $y$ vanish entirely:

Consistency

The equation $0=\lambda +\mu$ is the ultimate test of consistency.

If $\lambda +\mu =0$, the equation becomes $0=0$. This is a tautology, meaning the two equations represent the same line. Every point on one line is a point on the other, resulting in infinitely many solutions.

However, if $\lambda + \mu eq 0$, we are left with a contradiction: $0=\text{a non-zero constant}$. This is mathematically impossible, indicating that the lines are parallel but distinct. In this scenario, there is absolutely no solution.

The Mastery of Perspective

By analyzing the determinant, we found the boundary of the system's behavior. By testing the consistency condition, we unlocked the secrets of the parallel case.

You have successfully navigated the transition from algebraic manipulation to geometric intuition. Remember, in JEE Advanced, never just calculate—visualize. When you see a system of equations, see the lines, see the slopes, and always look for the consistency condition. You are now ready to tackle any linear system that comes your way!