Animated Solution for Mathematics - Matrices and Determinants: If the system of equations $3x+y+4z=3$, $2x+\alpha y-z=-3$, $x+2y+z=4$ has no solution, then the value of $\alpha$ is equal to :

We are investigating a system of three linear equations in three variables: $3x+y+4z=3$ $2x+\alpha y-z=-3$ $x+2y+z=4$

Our objective is to determine the value of $\alpha$ such that the system has no solution. Geometrically, this implies that the three planes do not share a common point of intersection.

Cramer's Rule

To identify the condition for inconsistency, we utilize the determinant of the coefficient matrix, denoted as $\Delta$. For a system to have no solution (or infinite solutions), the primary condition is that the determinant must vanish:

Expanding this determinant along the first row:

Calculating the $2\times 2$ minors:

Simplifying the expression:

This yields our candidate value: $\alpha =19$.

Why Verification Matters

A value of $\Delta =0$ is necessary for the system to be inconsistent, but it could also lead to infinite solutions. To ensure there is no solution, we must verify that at least one of the Cramer determinants (${\Delta }_x,{\Delta }_y,{\Delta }_z$) is non-zero.

Let us calculate ${\Delta }_x$ by replacing the first column with the constants $(3,-3,4)$:

Expanding ${\Delta }_x$:

Since $\Delta =0$ and $\Delta_x eq 0$, the system is inconsistent and possesses no solution. The required value is: