# Matrices and determinants: GATE CS question with solution

From the GATE Daily set of 27 October 2026, question 2 of 10 (https://gatedaily.in/q/2026-10-27-determinant-after-row-operations/). Difficulty: standard.

### Q.2 · Engineering Mathematics · Matrices and determinants · NAT, 2 marks (answer with a number)

Let $A$ be a real $3 \times 3$ matrix with $\det(A) = 6$. A matrix $B$ is obtained from $A$ by applying the following row operations in order:

1. interchange row 1 and row 3;
2. multiply row 2 by 3;
3. replace row 3 by (row 3 $+ \; 5 \times$ row 1).

The value of $\det\left(3\, B^{-1} A^{T}\right)$ is ________. (answer in integer)

**Answer:** -9

**Solution.** **Method.** Track $\det$ through each row operation, then use the multiplicative rules.

| Operation | Effect on det | Running value |
|---|---|---|
| start | | $6$ |
| swap rows 1 and 3 | sign flips | $-6$ |
| row 2 $\times 3$ | multiplied by 3 | $-18$ |
| row 3 $+ 5\times$ row 1 | unchanged | $-18$ |

So $\det(B) = -18$. For a $3 \times 3$ matrix, $\det(kM) = k^3 \det(M)$, $\det(B^{-1}) = 1/\det(B)$ and $\det(A^T) = \det(A)$. Hence

$$\det\left(3B^{-1}A^T\right) = 3^3 \cdot \frac{1}{\det B} \cdot \det A = 27 \cdot \frac{6}{-18} = -9.$$

**The trap.** Three slips each give a different wrong number: writing $\det(3M) = 3\det(M)$ instead of $3^3 \det(M)$ (gives $-1$); forgetting that a row swap flips the sign (gives $+9$); and thinking "add a multiple of another row" changes the determinant (it does not; only *scaling* a row does).

**What was tested.** The three elementary-row-operation rules for determinants and the identities $\det(kM) = k^n\det M$, $\det(M^{-1}) = 1/\det M$, $\det(M^T) = \det M$, $\det(MN) = \det M \det N$, all without expanding a single matrix.

**Remember:** Row swap flips the sign, scaling a row by c multiplies det by c, adding a multiple of one row to another changes nothing. Scaling the whole n x n matrix by k multiplies det by k^n, not k.