Official PYQ · 2023Q8Linear Algebra: Matrices, Determinants, System of Linear EquationsMCQ1 mark

GATE CS 2023 Q8 — Linear Algebra: Matrices, Determinants, System of Linear Equations

Let
$A=\left[\begin{array}{llll} 1 & 2 & 3 & 4 \\ 4 & 1 & 2 & 3 \\ 3 & 4 & 1 & 2 \\ 2 & 3 & 4 & 1 \end{array}\right]$
and
$B=\left[\begin{array}{llll} 3 & 4 & 1 & 2 \\ 4 & 1 & 2 & 3 \\ 1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 1 \end{array}\right]$
Let $\operatorname{det}(A)$ and $\operatorname{det}(B)$ denote the determinants of the matrices $A$ and $B,$ respectively.

Which one of the options given below is $\text{TRUE}?$

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Answer: B

To find the relationship between $\operatorname{det}(A)$ and $\operatorname{det}(B)$ , let us compare the rows of matrix $A$ and matrix $B$ .

Let the rows of matrix $A$ be denoted as $R_1, R_2, R_3,$ and $R_4$ respectively:

A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 4 & 1 & 2 & 3 \\ 3 & 4 & 1 & 2 \\ 2 & 3 & 4 & 1 \end{bmatrix} \begin{array}{l} \rightarrow R_1 \\ \rightarrow R_2 \\ \rightarrow R_3 \\ \rightarrow R_4 \end{array}

Now, observe the rows of matrix $B$ :

B = \begin{bmatrix} 3 & 4 & 1 & 2 \\ 4 & 1 & 2 & 3 \\ 1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 1 \end{bmatrix} \begin{array}{l} \rightarrow R_3 \\ \rightarrow R_2 \\ \rightarrow R_1 \\ \rightarrow R_4 \end{array}

We can see that matrix $B$ is obtained from matrix $A$ by swapping the first row ( $R_1$ ) and the third row ( $R_3$ ):

R_1 \longleftrightarrow R_3

By the properties of determinants, swapping any two rows of a matrix multiplies its determinant by $-1$ . Since exactly one row swap is performed to transform $A$ into $B$ :

\operatorname{det}(B) = -\operatorname{det}(A)

Thus, the correct option is B.

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Published 10 Oct 2026

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