NDA 2 2025 Maths, question 16

Topic: Matrices and determinants

Q16

The value of the determinant

$\left| \begin{matrix} a & b & c \\ l & m & n \\ p & q & r \end{matrix} \right|$

is equal to

(a) $\left| \begin{matrix} a & b & c \\ p & q & r \\ l & m & n \end{matrix} \right|$
(b) $\left| \begin{matrix} l & m & n \\ a & b & c \\ p & q & r \end{matrix} \right|$
(c) $\left| \begin{matrix} p & q & r \\ a & b & c \\ l & m & n \end{matrix} \right|$ Correct answer
(d) $\left| \begin{matrix} a & p & l \\ b & q & m \\ c & r & n \end{matrix} \right|$

Solution.

Answer: (c) $\left| \begin{matrix} p & q & r \\ a & b & c \\ l & m & n \end{matrix} \right|$

Option (c) cyclically permutes the rows from (1,2,3) to (3,1,2).
This is two row interchanges, so the determinant is unchanged.
Options (a) and (b) use one interchange; (d) is the transpose followed by one column interchange, and also changes the sign in general.