If the system of equations x + ay = 0, az + y = 0 and ax + z = 0 has infinite solutions, then the va

Q: Wait, why do we set the determinant to zero? Can't we just solve for $x, y, z$ directly?

That's a great instinct, but solving directly is like trying to catch smoke with your bare hands! When a system has infinite solutions, the variables are dependent on each other. Setting the determinant to zero is the 'master key'—it tells us exactly when the system loses its unique solution and becomes 'degenerate,' allowing for those infinite possibilities.

Q: Why do we only consider $a = -1$? What about the other roots of $a^3 = -1$?

Excellent catch! The equation $a^3 = -1$ actually has three roots in the complex plane: $-1$, $\frac{1 + i\sqrt{3}}{2}$, and $\frac{1 - i\sqrt{3}}{2}$. However, in the context of standard JEE problems involving geometric planes, we are almost always working in the real coordinate space $\mathbb{R}^3$. Thus, we restrict our search to real values of $a$.

Q: Is the trivial solution $(0,0,0)$ always included in the 'infinite solutions'?

Yes, absolutely! For any homogeneous system, the origin is always a solution. When we say 'infinite solutions,' we are saying that the origin is just one of many. It's like saying the origin is the 'anchor' point, and the line of intersection passes right through it.

Question

If the system of equations $x + ay = 0$, $az + y = 0$ and $ax + z = 0$ has infinite solutions, then the value of $a$ is

Options:

(A) (a) $-1$

(B) (b) 1

(C) (c) 0

(D) (d) no real values

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing in a vast, three-dimensional space. You have three planes, each defined by a simple linear equation: $x + ay = 0$, $az + y = 0$, and $ax + z = 0$.

Because these equations have no constant terms on the right side, they are what we call a **homogeneous system**. Geometrically, this means every single one of these planes passes directly through the origin $(0,0,0)$.

In a typical scenario, these three planes would intersect at a single point: the origin itself. This is the 'trivial solution' $(0,0,0)$.

However, the problem asks for the condition under which the system has **infinite solutions**. For this to happen, the planes must align in such a way that they intersect along an entire common line.

### The Determinant: The Gatekeeper of Solutions

To find when this system has infinite solutions, we need to look at the coefficient matrix $A$. We extract the coefficients of $x, y,$ and $z$ from our equations to build our matrix:

$$A = \begin{pmatrix} 1 & a & 0 \ 0 & 1 & a \ a & 0 & 1 \end{pmatrix}$$

For a homogeneous system to have non-trivial (infinite) solutions, the determinant of this matrix must be zero. The determinant measures the 'volume' spanned by the row vectors; if it is zero, the vectors are linearly dependent, meaning the planes are not independent.

We set $\Delta = |A| = 0$ and prepare for the expansion.

### The Algebraic Journey

Expanding the determinant along the first row is a straightforward process. We take the first element, $1$, and multiply it by the determinant of the remaining $2 	imes 2$ matrix, which is $(1 \cdot 1 - 0 \cdot a)$.

Then, we subtract the second element, $a$, multiplied by its minor, $(0 \cdot 1 - a \cdot a)$. The third term is $0$, so it vanishes:

$$\Delta = 1(1 - 0) - a(0 - a^2) + 0 = 0$$

Simplifying this, we get $1 - a(-a^2) = 0$, which leads us to the elegant equation:

$$1 + a^3 = 0$$

This is the heart of the problem. We are looking for the value of $a$ that satisfies $a^3 = -1$. Taking the real cube root, we find **$a = -1$**.

### Conclusion: The Beauty of Alignment

When **$a = -1$**, the system is no longer just three random planes; it becomes a perfectly aligned structure where the planes intersect along a common line.

We have successfully navigated the algebra and visualized the geometry. Remember, in JEE Advanced, it is rarely just about the calculation; it is about understanding the **why** behind the math.

You have just mastered the condition for infinite solutions in a homogeneous system. Keep this logic in your toolkit—it will serve you well in many more problems to come!

Animated Solution for Mathematics - Matrices and Determinants: If the system of equations $x + a y = 0$ , $a z + y = 0$ and $a x + z = 0$ has infinite solutions, then the value of $a$ is

Select Answer:

Visualized Solution

The System of Equations

Trivial vs Non-Trivial Solutions

Condition for Infinite Solutions

Constructing the Determinant

Expanding the Determinant

Simplifying the Expression

Forming the Polynomial

Solving for $a$

Final Conclusion

The Sigma Insight: Solution of System of Linear Equations (Matrix Method and Cramer's Rule)

Analyzing the Setup

The Determinant

The Algebraic Journey

Conclusion

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Animated Solution for Mathematics - Matrices and Determinants: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the system of equations x+ay=0, az+y=0 and ax+z=0 has infinite solutions, then the value of a is