Q.Evaluate $\begin{vmatrix} 1 & x & y \\ 1 & x+y & y \\ 1 & x & x+y \end{vmatrix}$.

Concept: Determinant Evaluation Using Identities (Row Operations)

Step 1: Apply $R_2 \to R_2 - R_1$ and $R_3 \to R_3 - R_1$:

$\begin{vmatrix} 1 & x & y \\ 0 & y & 0 \\ 0 & 0 & x \end{vmatrix}$ …

We evaluate

1. Row operations $R_2 \to R_2 - R_1$ and $R_3 \to R_3 - R_1$:

$\Delta = \begin{vmatrix} 1 & x & y \\ 0 & y & 0 \\ 0 & 0 & x \end{vmatrix}.$ …

Method: Row-Elimination to Reach a Near-Triangular Determinant

This method evaluates a determinant by using one row (often one with simple entries like 1's) to zero out the corresponding column in the other rows, avoiding a full cofactor expansion.

Steps

Step 1: Look for a row or column that repeats a simple value

Here, the first column is $(1,1,1)^T$ — three identical, simple entries. A repeated column entry is the cue to eliminate it using row operations rather than expanding directly.

Step 2: Subtract the "reference" row from the others

Using operation 3 ($R_i \to R_i + \lambda R_j$, value unchanged), subtract row 1 from every other row that shares the same first-column value:

This leaves the first column as $(1,0,0)^T$.

Step 3: Expand along the now-simple first column …

Common Mistakes

Mistake 1: Subtracting the wrong row (e.g. $R_1\to R_1-R_2$ instead of $R_2\to R_2-R_1$)

Why it's wrong: row operations must keep the row being modified on the left of the arrow — swapping which row is the "reference" changes every entry's sign and can silently flip the final answer's sign. Correct approach: always keep row 1 (with its 1's in the first column) fixed as the reference, and only modify rows 2 and 3.

Mistake 2: Expanding the full $3\times3$ before simplifying …