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NDA 2025 Maths · Mathematics · Matrices · medium

If a+b+c=0, then the determinant beginvmatrixa&b&c\b&c&a\c&a&bendvmatrix equals

Single correct option

If a+b+c=0a+b+c=0, then the determinant ∣abcbcacab∣\begin{vmatrix}a&b&c\\b&c&a\\c&a&b\end{vmatrix} equals
  1. A. abcabc
  2. B. 3abc3abc
  3. C. a3+b3+c3a^3+b^3+c^3
  4. D. 00
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Answer: (D) 00

Add rows 2 and 3 to row 1: row 1 becomes (a+b+c)(1,1,1)=(0,0,0)(a+b+c)(1,1,1)=(0,0,0). A determinant with a zero row is 00. In general the value is −(a3+b3+c3−3abc)-(a^3+b^3+c^3-3abc), which vanishes when a+b+c=0a+b+c=0.
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