GATE CS 2005 Q48 — Linear Algebra: Matrices, Determinants, System of Linear Equations

To determine the nature of the solutions for the given system of linear equations, we can analyze the determinant of the coefficient matrix $A$.

The system of equations is: $$\begin{align*} 2x_1 - x_2 + 3x_3 &= 1 \\ 3x_1 + 2x_2 + 5x_3 &= 2 \\ -x_1 + 4x_2 + x_3 &= 3 \end{align*}$$

The coefficient matrix $A$ is: $$A = \begin{pmatrix} 2 & -1 & 3 \\ 3 & 2 & 5 \\ -1 & 4 & 1 \end{pmatrix}$$

Let us calculate the determinant of $A$ (denoted as $\det(A)$): $$\begin{align*} \det(A) &= 2 \begin{vmatrix} 2 & 5 \\ 4 & 1 \end{vmatrix} - (-1) \begin{vmatrix} 3 & 5 \\ -1 & 1 \end{vmatrix} + 3 \begin{vmatrix} 3 & 2 \\ -1 & 4 \end{vmatrix} \\ &= 2(2 - 20) + 1(3 - (-5)) + 3(12 - (-2)) \\ &= 2(-18) + 1(8) + 3(14) \\ &= -36 + 8 + 42 \\ &= 14 \end{align*}$$

Since $\det(A) = 14 \neq 0$, the coefficient matrix $A$ is non-singular (invertible). This implies that the system has a unique solution given by $X = A^{-1}B$.