Animated Solution for Mathematics - Matrices and Determinants: Let the system of linear equations $x+2y+z=2$, $\alpha x+3y-z=\alpha$, $-\alpha x+y+2z=-\alpha$ be inconsistent. Then $\alpha$ is equal to :

Imagine you are standing in a 3D space, looking at three flat planes defined by our equations. Usually, these planes intersect at a single point, representing a unique solution $(x,y,z)$.

However, the system is inconsistent. This means there is no point in space where all three planes meet simultaneously. They might be parallel, or they might form a shape like a triangular tunnel, but they never lock together at a single coordinate.

Our mission is to find the value of $\alpha$ that forces this geometric impossibility.

Cramer's Rule

To solve this, we turn to the elegant machinery of Cramer's Rule. We define the coefficient matrix and its determinant, $\Delta$.

The rule states that for a system to have a unique solution, $\Delta$ must be non-zero. If $\Delta =0$, the system is either inconsistent or has infinite solutions.

To confirm inconsistency, we need $\Delta =0$ and at least one of the replacement determinants (${\Delta }_1,{\Delta }_2,\text{ or }{\Delta }_3$) to be non-zero. This is our mathematical litmus test.

Let's build our determinant $\Delta$ using the coefficients of $x,y,\text{ and }z$:

Expanding this along the first row, we get:

Simplifying this step-by-step:

This reduces beautifully to $\Delta =7-2\alpha +4\alpha$, which simplifies to $\Delta =7+2\alpha$. Setting $\Delta =0$ gives us $7+2\alpha =0$, leading to:

We must ensure this value of $\alpha$ doesn't lead to infinite solutions. We check ${\Delta }_1$ by replacing the first column of $\Delta$ with the constants $(2,\alpha ,-\alpha )$:

Expanding this, we find:

Substituting $\alpha =-\frac{7}{2}$ into ${\Delta }_1$, we get:

Since $7 eq 0$, the condition for inconsistency is satisfied. We have successfully navigated the algebra and confirmed that $\alpha =-\frac{7}{2}$ is indeed the value that breaks the system.