Let A = bmatrix 1 & 2 & 3 & 4 \\ 4 & 1 & 2 & 3 \\ 3 & 4 & 1 & 2 \\ 2 & 3 & 4 & 1 bmatrix and B = bmatrix 3 & 4 & 1 & 2 \\ 4 & 1 & 2 & 3 \\ 1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 1 bmatrix. Let (A) and (B) denote the determinants of the matrices A and B, respectively. Which one of the options given below is TRUE?

GATE CSE 2023 Q18: Relation between det(A) and det(B) for row-permuted matrices

Let

$A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 4 & 1 & 2 & 3 \\ 3 & 4 & 1 & 2 \\ 2 & 3 & 4 & 1 \end{bmatrix}$

and

$B = \begin{bmatrix} 3 & 4 & 1 & 2 \\ 4 & 1 & 2 & 3 \\ 1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 1 \end{bmatrix}.$

Let $\det(A)$ and $\det(B)$ denote the determinants of the matrices $A$ and $B$ , respectively.

Which one of the options given below is TRUE?

Try it: multiple choice: exactly one correct option

Wrong answer costs 1/3 mark in GATE.

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Answer: B (official GATE 2023 final answer key)

Concept used: Swapping two rows of a matrix changes the sign of its determinant.

First step: Compare the rows of $B$ with those of $A$ : $B$ is $A$ with rows 1 and 3 interchanged.

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Source: official GATE 2023 CS question paper and final answer key, organised by IIT Kanpur. Question paper · Answer key