If any two adjacent rows or columns of a determinant are interchanged in position, what will be the value of the determinant?

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Solution

Determine the value of determinant based on given information
Let us consider a $3 \times 3$ determinant
$A = |\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}|$
The value of determinant $A = a (e i - f h) - b (d i - f g) + c (d h - e g)$
Let us interchange the first and second rows of this determinant
$B = |\begin{matrix} d & e & f \\ a & b & c \\ g & h & i \end{matrix}|$
The value of this determinant $B = d (b i - c h) - e (a i - c g) + f (a h - b g) = - A$
Let us interchange the first and second columns of the determinant $A$
$C = |\begin{matrix} b & a & c \\ e & d & f \\ h & g & i \end{matrix}|$
The value of this determinant $C = b (d i - f g) - a (e i - f h) + c (e g - d h) = - A$
Hence, when two adjacent rows or columns of a determinant are interchanged, the value of the determinant changes its sign.

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