Among the several determinant properties in the NCERT Class 12 Determinants chapter, one specific result deals with what happens when the rows (or columns) of a determinant are shuffled in a cyclic order, rather than simply interchanged two at a time. This is subtly different from the familiar "swap two rows flips the sign" rule, and it is a genuinely useful shortcut on its own.
What "cyclic order" means
For a 3×3 determinant with rows R1,R2,R3, a cyclic rearrangement sends
R1→R2,R2→R3,R3→R1,
so the new determinant has its rows in the order R2,R3,R1. This moves every row at once, in the same rotational direction — it is not the same as interchanging just one pair of rows.
The property
Cyclically permuting the rows (or, equivalently, the columns) of a 3×3 determinant does not change its value.
Why it's true
A single row interchange flips the sign of a determinant. Sending R1 to the bottom while R2 and R3 shift up can be built from two successive interchanges: first swap R1↔R2 (sign flips once), then swap the row now in the second slot with R3 (sign flips again). Two sign flips cancel:
(−1)×(−1)=1.
So a full cyclic shift of three rows is an even permutation and leaves the determinant unchanged.
Note
This is specific to an odd count of rows. In general, cyclically permuting n rows carries a sign of (−1)n−1: for n=3 that's (−1)2=+1 (unchanged), but for n=4 rows a full cyclic shift does flip the sign, since (−1)3=−1.
A numeric check
Δ=1472583610=−3.
Cyclically shift the rows to R2,R3,R1:
Δ′=4715826103.
Expanding Δ′ along its first row: 4(8⋅3−10⋅2)−5(7⋅3−10⋅1)+6(7⋅2−8⋅1)=4(4)−5(11)+6(6)=16−55+36=−3 — the same value as Δ, confirming the cyclic shift left the determinant unchanged even though every row moved.
Watch out
Don't confuse this with a single row swap, which does flip the sign. The cyclic-invariance property only holds when all rows (or all columns) rotate together in one direction — shifting just one pair changes the value in the ordinary way.
Where this shows up: "cyclic determinants"
This property is the reason a whole family of board-exam "prove that" questions work, where the entries themselves are arranged cyclically, e.g. …