Q157 / 269 · NDA 2025 (Phase 1) · Mathematics

If $\Delta = \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix}$ and $A$, $B$, $C$, $D$, $G$ are the cofactors of the elements $a$, $b$, $c$, $d$, $g$ respectively, then what is $bB + cC - dD - gG$ equal to?

  • Correct answer
    A
    $0$
  • B
    $1$
  • C
    $\Delta$
  • D
    $-\Delta$

Solution

The sum of the products of elements of any row or column with their corresponding cofactors is equal to the determinant $\Delta$.

Expanding the determinant along the first row, we get:
$aA + bB + cC = \Delta$

Expanding the determinant along the first column, we get:
$aA + dD + gG = \Delta$

Subtracting the second equation from the first equation, we get:
$(aA + bB + cC) - (aA + dD + gG) = \Delta - \Delta$

$bB + cC - dD - gG = 0$

Answer: $0$

Hence, the correct answer is option: 'A'.

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