Animated Solution for Mathematics - Matrices and Determinants: For the system of linear equations $ax+y+z=1$, $x+ay+z=1$, $x+y+az=\beta$, which one of the following statements is NOT correct ?

Imagine you are standing in a three-dimensional room. You have three sheets of glass—three planes—defined by the equations:

Your goal is to find where these planes meet. In the world of JEE, this is not just algebra; it is geometry in motion.

The Determinant $\Delta$

Before we dive into the specific cases, we must consult the gatekeeper: the determinant of the coefficient matrix, $\Delta$. This value tells us everything about the nature of the intersection.

We construct our determinant from the coefficients of $x$, $y$, and $z$:

Expanding this carefully, we avoid common traps. Using the property of determinants, we add all rows to the first row, which reveals a common factor of $(\alpha +2)$. Factoring the remaining quadratic, we arrive at the elegant result:

This tells us that the system behaves uniquely whenever $\alpha eq 1$ and $\alpha eq -2$. When $\alpha$ hits these values, the system becomes singular, and the geometry changes.

Testing the Options

Now, we act as detectives, testing each scenario provided in the options.

For the case where $\alpha =2$ and $\beta =-1$, we calculate:

Since $\Delta eq 0$, the system must have a unique solution. If an option claims "infinitely many solutions" for these parameters, it is a direct contradiction.

For the case where $\alpha =2$ and $\beta =1$, we sum the equations:

This simplifies to $4(x+y+z)=3$, or $x+y+z=\frac{3}{4}$. This is perfectly consistent.

For the case where $\alpha =1$ and $\beta =1$, the equations collapse into $x+y+z=1$. All three planes are identical; they are the same plane. Thus, we have infinitely many solutions.

For the case where $\alpha =-2$ and $\beta =1$, the system becomes:

Summing these gives $0=3$, a classic contradiction. Therefore, no solution exists.

Mathematics is about patterns. By understanding the determinant $\Delta$, we didn't just solve a problem; we mapped the behavior of a system.

Whether it is a unique point, a line, or a plane, the algebra never lies. Keep practicing, keep visualizing, and remember: every equation tells a story.