GATE CS 2013 Q3 — Linear Algebra: Matrices, Determinants, System of Linear Equations

Let the given determinant be $D = \begin{vmatrix} 1 & x & x^{2}\\ 1& y & y^{2}\\ 1 & z & z^{2} \end{vmatrix}$. We can test the options using elementary column and row operations.

Testing Option A:

Apply the column operation $C_2 \leftarrow C_2 - C_3$:

Apply $C_2 \leftarrow C_2 + C_1$ followed by $C_3 \leftarrow C_3 - C_1$:

Interchanging columns $C_2$ and $C_3$ introduces a negative sign:

Since $D_A = -D$, Option A does not equal the original determinant $D$ (unless $D = 0$).

Verifying Option B:
Applying $C_2 \leftarrow C_2 - C_1$ and $C_3 \leftarrow C_3 - C_1$ on Option B directly yields $D$.

Verifying Option C:
Applying row operations $R_1 \leftarrow R_1 - R_2$ and $R_2 \leftarrow R_2 - R_3$ on the original determinant $D$ yields Option C, preserving the determinant value.