If \(\Delta = \begin{vmatrix}a & b & c\\ b & d & e\\ c & e & f\end{vmatrix}\), then the cofactor \(A_{32}\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(bc - ae\)
The cofactor is \(A_{ij} = (-1)^{i + j}M_{ij}\), where the minor \(M_{ij}\) is the determinant left after deleting row i and column j.
Deleting row 3 and column 2 leaves:
\(\displaystyle M_{32} = \begin{vmatrix}a & c\\ b & e\end{vmatrix} = ae - bc\)
\(A_{32} = (-1)^{5}(ae - bc)\)
\(A_{32} = bc - ae\)
Why the other options are incorrect
Option (1) \(ae - bc\) — Incorrect. This is the minor \(M_{32}\); the sign \((-1)^{3 + 2} = -1\) is missed.
Option (3) \(ae + bc\) — Incorrect. The minor is the difference \(ae - bc\), not a sum.
Option (4) \(-(ae + bc)\) — Incorrect. The sign is right, but the minor is \(ae - bc\), not \(ae + bc\).