Animated Solution for Mathematics - Matrices and Determinants: Consider the system of linear equations; $x_1+2x_2+x_3=3$, $2x_1+3x_2+x_3=3$, $3x_1+5x_2+2x_3=1$. The system has

We are presented with a system of three linear equations: $x_1+2x_2+x_3=3$ $2x_1+3x_2+x_3=3$ $3x_1+5x_2+2x_3=1$

Our objective is to classify the system as having a unique solution, infinite solutions, or no solution.

The Determinant $\Delta$

In linear algebra, the determinant of the coefficient matrix, denoted as $\Delta$, acts as the primary indicator of the system's nature. We construct the matrix using the coefficients of the variables:

To evaluate this, we expand along the first row:

Simplifying the terms inside the parentheses:

Since $\Delta =0$, the system does not possess a unique solution. We must now determine if the system is inconsistent or possesses infinitely many solutions.

Testing for Inconsistency

We utilize Cramer's Rule to investigate further. If $\Delta =0$ and at least one of the modified determinants (${\Delta }_1,{\Delta }_2,{\Delta }_3$) is non-zero, the system is inconsistent.

We calculate ${\Delta }_1$ by replacing the first column of the coefficient matrix with the constant terms $(3,3,1{)}^T$:

Expanding along the first row:

We have established that $\Delta =0$ and ${\Delta }_1=5$. Because $\Delta_1 eq 0$, the condition for inconsistency is strictly satisfied.

Geometrically, this implies that the three planes defined by the equations do not share a common intersection point or line. Therefore, the system has no solution.