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

Concept: Determinant Properties Cyclic — use row/column operations to exploit symmetry and simplify.

Step 1: Apply $C_1 \to C_1 + C_2 + C_3$. Each entry in $C_1$ becomes $x + y + (x+y) = 2(x+y)$:

Step 2: Factor $2(x+y)$ from $C_1$:

Step 3: Perform $R_2 \to R_2 - R_1$ and $R_3 \to R_3 - R_1$: …

We evaluate

1. Column operation $C_1 \to C_1 + C_2 + C_3$. Every row sum is $x + y + (x+y) = 2(x+y)$, so

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

3. Expand along the first column: …

Method: Column-Sum Simplification for Cyclic/Symmetric Determinants

This method solves determinants where each row (or column) sums to the same expression — a strong hint to use a column/row operation to factor that sum out before expanding.

Steps

Step 1: Check whether every row (or column) has the same sum

Add the entries across each row (or down each column). If the sum is identical for every row/column, that repeated value is a signal to combine columns (or rows) into one, since the corresponding operation leaves the determinant's value unchanged while revealing a common factor.

Step 2: Apply the combining operation $C_1 \to C_1 + C_2 + C_3$ (or the row analogue)

By operation 3 of the determinant row/column rules, adding one column's multiple to another doesn't change the determinant's value. Replacing $C_1$ by $C_1+C_2+C_3$ turns every entry of the first column into the same repeated sum $S$:

Step 3: Factor $S$ out of the column

Since every entry of $C_1$ is now $S$, factor it out using operation 2 (scaling a row/column pulls out a common multiple):

Step 4: Clear the remaining 1's with row operations, then expand …

Common Mistakes

Mistake 1: Forgetting the sign when subtracting rows after factoring out $2(x+y)$

Why it's wrong: after $C_1\to C_1+C_2+C_3$ and factoring, students often apply $R_2\to R_2-R_1$ but subtract entry-by-entry inconsistently, producing a row that doesn't actually have a zero in the first position. Correct approach: subtract the entire row 1 from row 2 (and row 3) term by term, one column at a time, and double-check the first column really becomes $0$ before moving on.

Mistake 2: Missing that all three rows share the same sum, and expanding the original $3\times3$ directly …