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Tuesday27 October 2026

GATE CS & IT · 2027 · Ten questions a day

Past sets27 Oct 2026Maths

Matrices and determinants: GATE CS question with solution

Q.2

Let be a real matrix with . A matrix is obtained from 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 row 1).

The value of is ________. (answer in integer)

One try. It counts in your 27 Oct 2026 set.
Show solution
RememberRow 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.

Solution NAT · 2 marks · no negative marking

Answer: -9

Method. Track through each row operation, then use the multiplicative rules.

Operation Effect on det Running value
start
swap rows 1 and 3 sign flips
row 2 multiplied by 3
row 3 row 1 unchanged

So . For a matrix, , and . Hence

The trap. Three slips each give a different wrong number: writing instead of (gives ); forgetting that a row swap flips the sign (gives ); 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 , , , , all without expanding a single matrix.

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